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Quadratic Equations

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A quadratic equation is a mathematical expression where the highest power of the variable is two. In simple terms, the variable (usually 'x') is squared. The standard form of this equation is ax² + bx + c = 0. Here, 'a', 'b', and 'c' are known numbers called coefficients. The most important rule is that 'a' cannot be zero. If 'a' were zero, the 'x squared' term would disappear. Then it would just be a simple linear equation.

Concepts (7)

Quickly compare roots of two quadratic equations (x and y) using factorization and sign analysis. Prioritize shortcuts to determine x > y, x < y, x = y, or no relation.

To compare roots of two quadratic equations, typically given as ax² + bx + c = 0 for 'x' and py² + qy + r = 0 for 'y', the goal is to find the relationship between 'x' and 'y'. This involves finding the roots of both equations and then comparing all possible pairs.

Core Formula (Factorization Method)

For a quadratic equation ax² + bx + c = 0, find two numbers p and q such that p + q = b and p * q = a * c. The equation can then be factored as (ax + p)(x + q/a) = 0 or more commonly, by splitting the middle term: ax² + px + qx + c = 0. The roots will be -p/a and -q/a (or the values that make the factors zero).

Worked Example 1

Compare x and y: I. x² - 13x + 42 = 0 II. y² - 15y + 56 = 0

Solution: For I: x² - 13x + 42 = 0 We need two numbers that multiply to 42 and add up to -13. These are -6 and -7. (x - 6)(x - 7) = 0 Roots for x are: x = 6, 7

For II: y² - 15y + 56 = 0 We need two numbers that multiply to 56 and add up to -15. These are -7 and -8. (y - 7)(y - 8) = 0 Roots for y are: y = 7, 8

Comparison:

  • Compare x=6 with y=7 -> 6 < 7
  • Compare x=6 with y=8 -> 6 < 8
  • Compare x=7 with y=7 -> 7 = 7
  • Compare x=7 with y=8 -> 7 < 8

Since all values of x are either less than or equal to values of y, the relation is x ≤ y.

Worked Example 2

Compare x and y: I. 2x² + 9x + 9 = 0 II. 2y² + 17y + 36 = 0

Solution: For I: 2x² + 9x + 9 = 0 a*c = 2*9 = 18. We need two numbers that multiply to 18 and add up to 9. These are 3 and 6. 2x² + 3x + 6x + 9 = 0 x(2x + 3) + 3(2x + 3) = 0 (x + 3)(2x + 3) = 0 Roots for x are: x = -3, -3/2 = -1.5

For II: 2y² + 17y + 36 = 0 a*c = 2*36 = 72. We need two numbers that multiply to 72 and add up to 17. These are 8 and 9. 2y² + 8y + 9y + 36 = 0 2y(y + 4) + 9(y + 4) = 0 (2y + 9)(y + 4) = 0 Roots for y are: y = -9/2 = -4.5, -4

Comparison:

  • Compare x=-3 with y=-4.5 -> -3 > -4.5
  • Compare x=-3 with y=-4 -> -3 > -4
  • Compare x=-1.5 with y=-4.5 -> -1.5 > -4.5
  • Compare x=-1.5 with y=-4 -> -1.5 > -4

Since all values of x are greater than all values of y, the relation is x > y.

Shortcuts & Tricks

1. Sign Analysis (Most Important for Speed): Quickly determine the sign of the roots based on the coefficients b and c (assuming a is positive. If a is negative, multiply the whole equation by -1).

Equation Form (ax² + bx + c = 0)Signs of Roots
+ + + (e.g., x² + 5x + 6 = 0)Both Negative (-, -)
- + + (e.g., x² - 5x + 6 = 0)Both Positive (+, +)
+ + - (e.g., x² + x - 6 = 0)One Positive, One Negative (Larger magnitude root is -)
- + - (e.g., x² - x - 6 = 0)One Positive, One Negative (Larger magnitude root is +)

How to use:

  • If Eq1 has + roots and Eq2 has - roots, then x > y directly. No calculation needed.
  • If Eq1 has +,- roots and Eq2 has +,- roots, then 'No Relation' is highly probable. Calculate only if necessary.
  • If both equations have roots of the same sign (e.g., both +, + or both -, -), then you must calculate the exact values for comparison.

2. Coefficient Comparison (Rare but useful): If the equations are very similar, sometimes you can infer the relation without full calculation. E.g., x² - 5x + 6 = 0 (roots 2,3) vs x² - 6x + 8 = 0 (roots 2,4). Here, x ≤ y is not immediately obvious without calculation, but if the constants are very different, it might be.

Common Mistakes

  1. Incorrect Sign Application: A common error is misapplying signs during factorization or when determining the final roots. Remember, if (x-a)(x-b)=0, then x=a and x=b.
  2. Incomplete Comparison: Not comparing ALL roots of x with ALL roots of y. If even one pair violates a relation (e.g., x1 > y1 but x2 < y2), then the answer is 'No Relation' or 'Cannot be determined'.
  3. Ignoring 'a' in ax²: When a ≠ 1, remember to divide the factors by a to get the actual roots, or use the p/a, q/a method. E.g., for 2x² + 5x + 3 = 0, factors are (2x+3)(x+1)=0, roots are -3/2 and -1.

Derivation (Brief)

The sign analysis shortcut stems directly from Vieta's formulas and the quadratic formula x = (-b ± √(b²-4ac)) / 2a.

  1. Product of Roots (c/a): If c/a is positive, both roots must have the same sign. If c/a is negative, roots must have opposite signs.
  2. Sum of Roots (-b/a): If c/a is positive (same sign roots):
    • If -b/a is positive, both roots are positive.
    • If -b/a is negative, both roots are negative.
  3. If c/a is negative (opposite sign roots), the sign of the larger magnitude root depends on -b/a:
    • If -b/a is positive, the positive root has a larger magnitude.
    • If -b/a is negative, the negative root has a larger magnitude.

This logic forms the basis of the sign analysis table, allowing rapid elimination of options without full calculation.

Advanced Examples

Example: I. x² - 20x + 96 = 0 II. y² - 25y + 156 = 0

Solution: Both equations are '- + +' form, meaning both have +ve, +ve roots. Calculation is required. For I: x² - 20x + 96 = 0 -> Factors of 96 that sum to 20 are 8 and 12. So x = 8, 12. For II: y² - 25y + 156 = 0 -> Factors of 156 that sum to 25 are 12 and 13. So y = 12, 13.

Comparison:

  • x=8 vs y=12 -> 8 < 12
  • x=8 vs y=13 -> 8 < 13
  • x=12 vs y=12 -> 12 = 12
  • x=12 vs y=13 -> 12 < 13

Result: x ≤ y

Variation Types

Sometimes, one equation might be linear or involve square roots/cubes. For SSC CGL, the primary focus remains on two quadratic equations. However, be prepared for:

  1. One Linear, One Quadratic: Solve the linear equation directly for one variable, then substitute into the quadratic or compare directly. E.g., x - 3 = 0 and y² - 5y + 6 = 0.
  2. Equations with Square/Cube Roots: E.g., x = √16 (x=4) or y³ = 64 (y=4). Remember √16 only gives +4, but x²=16 gives x=±4.

Time-Saving Methods

  1. Prioritize Sign Analysis: Always start with this. It's the fastest way to eliminate options or directly find the relation (especially 'No Relation').
  2. Mental Math for Factorization: Practice finding factors quickly. For ax² + bx + c = 0, look for factors of a*c that sum to b.
  3. Approximate Calculation: If coefficients are large, sometimes you don't need exact values. For example, if x roots are +10, +12 and y roots are +2, +3, you know x > y without precise calculation if the numbers are clearly separated.
  4. Stop Early: If, during comparison, you find x1 > y1 and then x2 < y2, you can immediately conclude 'No Relation' without comparing all remaining pairs.
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Master quadratic equations for SSC CGL by quickly applying factorization or the quadratic formula. Focus on roots, discriminant, and coefficient relations for speed and accuracy.

Quadratic equations are critical for SSC CGL, appearing directly and in word problems. The standard form is ax² + bx + c = 0, where 'a' ≠ 0. Every quadratic equation has exactly two roots (solutions).

Core Formula

There are two primary methods to solve quadratic equations:

  1. Factorization Method: If you can express the quadratic as a product of two linear factors, say (px + q)(rx + s) = 0, then the roots are found by setting each factor to zero: px + q = 0 and rx + s = 0.

    • Key Idea: Find two numbers whose product is ac and whose sum is b.

  2. Quadratic Formula: This formula works for all quadratic equations. x = [-b ± sqrt(b² - 4ac)] / 2a

    • Discriminant (D): The term b² - 4ac is called the discriminant. It tells us about the nature of the roots:
      • If D > 0: Two distinct real roots.
      • If D = 0: Two equal real roots (a repeated root).
      • If D < 0: Two distinct complex (imaginary) roots.

Worked Example 1 (Factorization)

Problem: Solve x² - 7x + 12 = 0 Solution:

  1. Identify a=1, b=-7, c=12.
  2. Find two numbers whose product is ac = 1 * 12 = 12 and whose sum is b = -7. These numbers are -3 and -4.
  3. Rewrite the middle term: x² - 3x - 4x + 12 = 0
  4. Factor by grouping: x(x - 3) - 4(x - 3) = 0
  5. Factor out the common term: (x - 3)(x - 4) = 0
  6. Set each factor to zero: x - 3 = 0 or x - 4 = 0
  7. Roots are x = 3 and x = 4.

Worked Example 2 (Quadratic Formula)

Problem: Solve 2x² + 5x - 3 = 0 Solution:

  1. Identify a=2, b=5, c=-3.
  2. Calculate the discriminant D = b² - 4ac = (5)² - 4(2)(-3) = 25 - (-24) = 25 + 24 = 49.
  3. Since D = 49 > 0, there are two distinct real roots.
  4. Apply the quadratic formula: x = [-b ± sqrt(D)] / 2a x = [-5 ± sqrt(49)] / (2 * 2) x = [-5 ± 7] / 4
  5. Calculate the two roots: x1 = (-5 + 7) / 4 = 2 / 4 = 1/2 x2 = (-5 - 7) / 4 = -12 / 4 = -3
  6. Roots are x = 1/2 and x = -3.

Shortcuts & Tricks

  1. Sum and Product of Roots: For ax² + bx + c = 0, if roots are α and β:
    • Sum of roots: α + β = -b/a
    • Product of roots: α * β = c/a
    • Trick: If a=1, then x² - (sum)x + (product) = 0. This is very fast for a=1 cases.
  2. Checking Options: In MCQ exams, if the roots are given in options, substitute them into the equation. The option that satisfies the equation is correct. This is often faster than solving from scratch.
  3. Special Cases for Roots:
    • If a + b + c = 0, then one root is 1, and the other root is c/a.
    • If a - b + c = 0, then one root is -1, and the other root is -c/a.

Common Mistakes

  1. Sign Errors: Misplacing negative signs, especially when b or c are negative, is a frequent error in the quadratic formula and factorization.
  2. Incorrect Discriminant Calculation: Forgetting 4ac or miscalculating b² (e.g., (-3)² becoming -9 instead of 9).
  3. Dividing by a: In the quadratic formula, ensure the entire numerator [-b ± sqrt(D)] is divided by 2a, not just part of it.

Derivation (Brief)

The quadratic formula can be derived by the method of 'completing the square'. Starting with ax² + bx + c = 0:

  1. Divide by a: x² + (b/a)x + (c/a) = 0
  2. Move constant term: x² + (b/a)x = -c/a
  3. Complete the square on the left by adding (b/2a)² to both sides: x² + (b/a)x + (b/2a)² = -c/a + (b/2a)²
  4. Factor the left side and simplify the right: (x + b/2a)² = (b² - 4ac) / 4a²
  5. Take the square root of both sides: x + b/2a = ±sqrt(b² - 4ac) / 2a
  6. Isolate x: x = -b/2a ± sqrt(b² - 4ac) / 2a
  7. Combine terms: x = [-b ± sqrt(b² - 4ac)] / 2a

Advanced Examples

Problem: Solve (x+2)² - 5(x+2) + 6 = 0 Solution:

  1. This is an equation reducible to quadratic form. Let y = x+2.
  2. Substitute y into the equation: y² - 5y + 6 = 0
  3. Factorize: (y - 2)(y - 3) = 0
  4. Roots for y are y = 2 and y = 3.
  5. Substitute back x+2 for y:
    • x+2 = 2 => x = 0
    • x+2 = 3 => x = 1
  6. Roots are x = 0 and x = 1.

Variation Types

  1. Equations with fractions: E.g., x + 1/x = 5. Multiply by x to get x² + 1 = 5x, which rearranges to x² - 5x + 1 = 0.
  2. Equations with higher powers reducible to quadratic: E.g., x⁴ - 5x² + 4 = 0. Let y = x². Then y² - 5y + 4 = 0. Solve for y, then find x from x = ±sqrt(y).

Time-Saving Methods

  1. Quick Factorization for a=1: For x² + bx + c = 0, directly look for two numbers p, q such that p+q=b and p*q=c. The roots are -p and -q. (Note: If x² - (p+q)x + pq = 0, roots are p, q). Be careful with signs.
    • Example: x² - 5x + 6 = 0. Numbers are -2, -3. So x = 2, 3.
  2. Nature of Roots First: Before solving, quickly calculate D = b² - 4ac. If D < 0, you know there are no real roots, which can save time if the question asks for real roots or if options only contain real numbers.
  3. Relationship between roots and coefficients: If a question asks for α² + β² or 1/α + 1/β, use α + β = -b/a and αβ = c/a.
    • α² + β² = (α + β)² - 2αβ
    • 1/α + 1/β = (α + β) / αβ This avoids calculating individual roots first.
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Factorization splits quadratic expressions into simpler linear factors, crucial for quickly finding roots. Master middle term splitting for SSC CGL speed and accuracy.

Factorization of quadratic expressions, typically in the form ax² + bx + c, is a fundamental skill for SSC CGL. It allows you to break down a complex expression into a product of simpler linear factors, making it easy to find the roots (values of x that make the expression zero) using the zero-product rule.

Core Formula

The primary method for factorization is Middle Term Splitting. For a quadratic expression ax² + bx + c:

  1. Find the product ac.
  2. Find two numbers, say p and q, such that p × q = ac and p + q = b.
  3. Rewrite the middle term bx as px + qx.
  4. Group the terms and factor out common monomials.

Worked Example 1

Factorize: x² + 7x + 12

  1. Here, a=1, b=7, c=12. Product ac = 1 × 12 = 12.
  2. We need two numbers that multiply to 12 and add to 7. These are 3 and 4 (3 × 4 = 12, 3 + 4 = 7).
  3. Rewrite 7x as 3x + 4x: x² + 3x + 4x + 12
  4. Group and factor: x(x + 3) + 4(x + 3) (x + 3)(x + 4)

Worked Example 2

Factorize: 3x² - 10x + 8

  1. Here, a=3, b=-10, c=8. Product ac = 3 × 8 = 24.
  2. We need two numbers that multiply to 24 and add to -10. These are -6 and -4 ((-6) × (-4) = 24, (-6) + (-4) = -10).
  3. Rewrite -10x as -6x - 4x: 3x² - 6x - 4x + 8
  4. Group and factor: 3x(x - 2) - 4(x - 2) (3x - 4)(x - 2)

Shortcuts & Tricks

  • For x² + bx + c (when a=1): Directly find two numbers that multiply to c and add to b. If these numbers are p and q, the factors are (x+p)(x+q). This is the fastest method.
  • Option Elimination: If multiple-choice options are given, substitute the roots from the options back into the original equation. The option that makes the expression zero is correct. This can be faster than full factorization for complex expressions.
  • Difference of Squares: Recognize a² - b² = (a - b)(a + b) instantly. E.g., x² - 25 = (x - 5)(x + 5).

Common Mistakes

  1. Sign Errors: Incorrectly handling positive/negative signs when finding p and q for ac and b. Always double-check p × q = ac and p + q = b with signs.
  2. Incorrect Grouping: After splitting the middle term, students sometimes group terms incorrectly or fail to factor out the common binomial term.
  3. Forgetting Common Factors: Always check if there's a common factor in all terms of the quadratic expression before applying middle term splitting. E.g., 2x² + 10x + 12 should first be 2(x² + 5x + 6).

Derivation (brief)

Factorization relies on the Zero Product Rule: If the product of two or more factors is zero, then at least one of the factors must be zero. When we factorize ax² + bx + c = 0 into (dx + e)(fx + g) = 0, we can then set each factor to zero: dx + e = 0 or fx + g = 0, which directly gives us the roots x = -e/d and x = -g/f. The middle term splitting method essentially reverses the FOIL (First, Outer, Inner, Last) multiplication process used to expand (dx + e)(fx + g).

Advanced Examples

1. Factorize: 6x² + 5x - 6

  • a=6, b=5, c=-6. Product ac = 6 × (-6) = -36.
  • Need two numbers that multiply to -36 and add to 5. These are 9 and -4.
  • 6x² + 9x - 4x - 6
  • 3x(2x + 3) - 2(2x + 3)
  • (3x - 2)(2x + 3)

2. Factorize: 4x² - 12x + 9 (Recognize as a perfect square)

  • This is of the form (Ax - B)² = A²x² - 2ABx + B².
  • Here, A² = 4, so A = 2. B² = 9, so B = 3.
  • Check middle term: -2ABx = -2(2)(3)x = -12x. This matches.
  • So, (2x - 3)² or (2x - 3)(2x - 3).

Variation Types

  • Difference of Squares: x² - y² = (x - y)(x + y). Very common. E.g., 49x² - 81 = (7x - 9)(7x + 9).
  • Perfect Square Trinomials: a² + 2ab + b² = (a + b)² and a² - 2ab + b² = (a - b)². Recognizing these saves significant time.
  • Common Monomial Factors: Always look for a common factor first. E.g., 5x³ - 20x can be factored as 5x(x² - 4), then 5x(x - 2)(x + 2). This simplifies the quadratic part before applying other methods.

Time-Saving Methods

  • Reverse Check with Options: For problems asking for factors or roots, if options are provided, pick an option and multiply the factors or substitute the roots into the original equation. This is often the quickest way to verify the correct answer, especially for complex coefficients.
  • Mental Math for a=1: Practice finding factors of c that sum to b mentally. This is a huge time-saver for x² + bx + c type problems.
  • Pattern Recognition: Train your eye to spot difference of squares and perfect square trinomials immediately. This avoids the longer middle term splitting process for these specific types.
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Master quadratic roots and nature using discriminant (D=b²-4ac). D>0 (real, distinct), D=0 (real, equal), D<0 (complex). Quickly find sum (-b/a) and product (c/a) of roots for exam speed.

Core Formulas

For a standard quadratic equation ax² + bx + c = 0 (where a ≠ 0):

  1. Discriminant (D): The discriminant determines the nature of the roots. D = b² - 4ac

  2. Nature of Roots:

    • If D > 0: Roots are Real and Distinct (different).
      • If D is a perfect square, roots are Rational.
      • If D is not a perfect square, roots are Irrational.
    • If D = 0: Roots are Real and Equal (identical). Each root is -b/2a.
    • If D < 0: Roots are Imaginary/Complex and Distinct (conjugate pairs).

  3. Sum of Roots (α + β): If α and β are the roots, their sum is: α + β = -b/a

  4. Product of Roots (αβ): The product of the roots is: αβ = c/a

  5. Forming a Quadratic Equation: If roots α and β are given, the equation is: x² - (α + β)x + αβ = 0

Worked Example 1

Q: For the equation 3x² - 7x + 2 = 0, find the nature of its roots, their sum, and their product.

Solution: Here, a = 3, b = -7, c = 2.

  1. Discriminant (D): D = b² - 4ac = (-7)² - 4(3)(2) = 49 - 24 = 25
  2. Nature of Roots: Since D = 25 > 0 and 25 is a perfect square, the roots are Real, Distinct, and Rational.
  3. Sum of Roots: α + β = -b/a = -(-7)/3 = 7/3
  4. Product of Roots: αβ = c/a = 2/3

Worked Example 2

Q: Find the value of k for which the roots of the equation x² + (k-2)x + 9 = 0 are equal.

Solution: For roots to be equal, the discriminant D must be 0. Here, a = 1, b = (k-2), c = 9. D = b² - 4ac = 0 (k-2)² - 4(1)(9) = 0 (k-2)² - 36 = 0 (k-2)² = 36 k-2 = ±√36 k-2 = ±6 Two possibilities:

  1. k-2 = 6 => k = 8
  2. k-2 = -6 => k = -4 Thus, k = 8 or k = -4.

Shortcuts & Tricks

  1. Quick Check for Rational Roots: If D is a perfect square, roots are rational. This saves time from actually calculating the roots.
  2. Sum/Product for Symmetric Expressions: If asked for α² + β² or 1/α + 1/β, use α + β and αβ directly. For example, α² + β² = (α + β)² - 2αβ.
  3. Coefficient Sum: If a + b + c = 0, then one root is 1 and the other is c/a. (e.g., x² - 5x + 4 = 0 -> 1 - 5 + 4 = 0, so roots are 1 and 4/1 = 4).
  4. Coefficient Alternating Sum: If a - b + c = 0, then one root is -1 and the other is -c/a. (e.g., x² + 5x + 4 = 0 -> 1 - 5 + 4 = 0, so roots are -1 and -4/1 = -4).

Common Mistakes

  1. Sign Errors in Discriminant: Forgetting that b or c can be negative, leading to incorrect b² - 4ac calculation. Always use parentheses: (-b)².
  2. Confusing Sum and Product Formulas: Mixing up -b/a and c/a. Remember sum has the negative sign.
  3. Misinterpreting D=0: Students sometimes confuse D=0 (equal roots) with D>0 (distinct roots). Equal roots mean only one unique solution value, repeated twice.

Derivation (brief)

The quadratic formula, which gives the roots of ax² + bx + c = 0 as x = [-b ± √(b² - 4ac)] / 2a, is derived by the method of completing the square. The term b² - 4ac under the square root is the discriminant. Its value dictates whether the square root yields a real number (if non-negative) or an imaginary number (if negative), thus determining the nature of the roots.

Advanced Examples

Q: If one root of 2x² - 8x + k = 0 is three times the other, find the value of k.

Solution: Let the roots be α and 3α. From the equation, a=2, b=-8, c=k. Sum of roots: α + 3α = -b/a => 4α = -(-8)/2 => 4α = 4 => α = 1. Product of roots: α * 3α = c/a => 3α² = k/2. Substitute α = 1: 3(1)² = k/2 => 3 = k/2 => k = 6.

Q: If the roots of px² + qx + r = 0 are reciprocal to each other, what is the relation between p, q, and r?

Solution: Let the roots be α and 1/α. Product of roots: α * (1/α) = c/a 1 = r/p So, p = r.

Variation Types

  1. Equations Reducible to Quadratic Form: Equations like x⁴ - 5x² + 4 = 0 can be solved by substituting y = x², transforming it into y² - 5y + 4 = 0.
  2. Common Roots: Problems involving two quadratic equations having one or both roots in common. For one common root, use the condition that the root satisfies both equations.
  3. Transformation of Roots: If α, β are roots of ax² + bx + c = 0, find an equation whose roots are α+k, β+k or 1/α, 1/β, etc. Use sum and product of new roots in x² - (Sum)x + (Product) = 0.

Time-Saving Methods

  • Vieta's Formulas: The sum and product of roots are direct applications of Vieta's formulas. For higher-degree polynomials, these formulas extend to sums of roots taken n at a time.
  • Graphical Interpretation: A quadratic equation ax² + bx + c = 0 represents a parabola. The roots are the x-intercepts. If D>0, it crosses the x-axis at two points. If D=0, it touches the x-axis at one point. If D<0, it doesn't intersect the x-axis.
  • Trial and Error for Integer Roots: For simple equations, if c/a has few factors, try substituting them to quickly find integer roots before resorting to the formula.
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The nature of roots depends on the Discriminant (D = b² - 4ac). If D is greater than zero, the roots are real and different. If D is equal to zero, the roots are real and exactly the same. If D is less than zero, the roots are not real (imaginary).

The nature of roots depends on the Discriminant (D = b² - 4ac). If D is greater than zero, the roots are real and different. If D is equal to zero, the roots are real and exactly the same. If D is less than zero, the roots are not real (imaginary). For example, in x² - 6x + 9 = 0, D is (36 - 36) = 0. Thus, both roots are the same (3 and 3).

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You can find the sum and product of roots without solving the whole equation. Let the roots be α and β. The sum (α + β) is -b/a. The product (αβ) is c/a. This is very useful for verifying your answers or solving 'value of k' questions.

You can find the sum and product of roots without solving the whole equation. Let the roots be α and β. The sum (α + β) is -b/a. The product (αβ) is c/a. This is very useful for verifying your answers or solving 'value of k' questions. Example: In 2x² - 8x + 6 = 0, the sum of roots is -(-8)/2 = 4, and the product is 6/2 = 3.

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When you cannot break the equation into simple factors, use the universal Quadratic Formula. It calculates the two values of x directly using the coefficients a, b, and c. The 'plus-minus' sign in the formula creates the two distinct roots.

When you cannot break the equation into simple factors, use the universal Quadratic Formula. It calculates the two values of x directly using the coefficients a, b, and c. The 'plus-minus' sign in the formula creates the two distinct roots. For example, in x² - 5x + 6 = 0, using the formula gives x = 3 and x = 2.

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Start Lesson: Comparing Roots