Quadratic Formula: Solve Equations

Grade 9 · algebra · 101 practice problems · read aloud

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The Quadratic Formula 🎯

What is it and why is it useful?

The quadratic formula is a powerful tool that gives you the solutions (or "roots") for any quadratic equation. A quadratic equation is an equation that can be written in the standard form: ax² + bx + c = 0.

It's incredibly useful because it always works, even when factoring is difficult or impossible. Whenever you're stuck, the quadratic formula is your reliable backup plan!

The Formula

The famous formula is:

x = [-b ± √(b² - 4ac)] / 2a

Step-by-Step Guide

  1. Identify a, b, and c from your equation: ax² + bx + c = 0.
  2. Write down the formula so you don't forget it.
  3. Substitute the values of a, b, and c directly into the formula.
  4. Simplify under the radical (the b² - 4ac part). This is called the discriminant.
  5. Simplify the entire expression, including the square root and the fraction.
  6. Write both solutions: one with the '+' sign and one with the '-' sign.

Worked Examples

Example 1: Solve x² + 6x + 8 = 0

  1. a = 1, b = 6, c = 8
  2. x = [-6 ± √(6² - 4(1)(8))] / 2(1)
  3. x = [-6 ± √(36 - 32)] / 2
  4. x = [-6 ± √4] / 2
  5. x = [-6 ± 2] / 2
  6. Solutions: x = (-6+2)/2 = -2 and x = (-6-2)/2 = -4

Example 2: Solve 2x² - 4x - 6 = 0

  1. a = 2, b = -4, c = -6
  2. x = [4 ± √((-4)² - 4(2)(-6))] / 2(2)
  3. x = [4 ± √(16 + 48)] / 4
  4. x = [4 ± √64] / 4
  5. x = [4 ± 8] / 4
  6. Solutions: x = (4+8)/4 = 3 and x = (4-8)/4 = -1

Common Mistakes to Avoid 🚫

  • Forgetting the ± sign: You MUST write both the plus AND minus versions.
  • Sign errors with b: If b is negative, -b becomes positive. For -b, write the opposite sign.
  • Not using parentheses: When substituting, put values in parentheses to avoid order of operations errors.
  • Only dividing part of the numerator: The entire numerator (both terms!) is divided by 2a.

Tips & Tricks

  • Memory Aid: Sing to "Pop Goes the Weasel": "x equals negative b, plus or minus the square root, of b squared minus four a c, all over two a!"
  • Check the Discriminant: The part under the radical (b² - 4ac) tells you about the solutions:
    • Positive = Two real solutions
    • Zero = One real solution
    • Negative = Two complex solutions
  • Simplify first: Always check if you can divide the entire equation by a common factor to make numbers smaller.

How to Practice

  • Start with equations where a=1 to build confidence.
  • Practice with equations that have negative b and c values.
  • Mix it up! Try some problems that are factorable and some that aren't.
  • Always check your answers by plugging them back into the original equation.
  • Use online practice sites like Khan Academy for instant feedback.

Practice problems

6 of the 101, worked through step by step — try them before opening the answer.

1 7x² - 9x - 8 = 0

Hint: Identify the coefficients a, b, and c from the standard form ax² + bx + c = 0, then substitute them into the quadratic formula.

Show the answer

Answer: x = (9 ± √(193))/14

  1. Identify coefficients: a = 7, b = -9, c = -8
  2. Apply quadratic formula: x = [-b ± √(b² - 4ac)] / (2a)
  3. Substitute values: x = [9 ± √((-9)² - 4(7)(-8))] / (2×7)
  4. Calculate discriminant: (-9)² - 4(7)(-8) = 81 - (-224) = 81 + 224 = 305
  5. Simplify: x = [9 ± √305] / 14
  6. Final answer: x = (9 ± √305)/14

2 4x² + 7x - 2 = 0

Hint: Identify the coefficients a, b, and c from the standard quadratic form ax² + bx + c = 0, then substitute them into the quadratic formula.

Show the answer

Answer: x = 0.25, x = -2

  1. Identify coefficients: a = 4, b = 7, c = -2
  2. Apply quadratic formula: x = [-b ± √(b² - 4ac)] / (2a)
  3. Calculate discriminant: b² - 4ac = 7² - 4(4)(-2) = 49 + 32 = 81
  4. Find square root of discriminant: √81 = 9
  5. Substitute into formula: x = [-7 ± 9] / (2×4) = [-7 ± 9] / 8
  6. Calculate first solution: x = (-7 + 9)/8 = 2/8 = 0.25
  7. Calculate second solution: x = (-7 - 9)/8 = -16/8 = -2
  8. Final solutions: x = 0.25 and x = -2

3 3x² - 11x + 6 = 0

Hint: Identify the coefficients a, b, and c from the quadratic equation ax² + bx + c = 0, then substitute them into the quadratic formula x = [-b ± √(b² - 4ac)] / (2a). Remember to simplify the square root and the fraction.

Show the answer

Answer: x = 3, x = 2/3

  1. Identify coefficients: a = 3, b = -11, c = 6
  2. Calculate the discriminant: b² - 4ac = (-11)² - 4(3)(6) = 121 - 72 = 49
  3. Apply quadratic formula: x = [11 ± √49] / (2×3) = [11 ± 7] / 6
  4. Solve for both cases: Case 1: x = (11 + 7) / 6 = 18 / 6 = 3 Case 2: x = (11 - 7) / 6 = 4 / 6 = 2/3
  5. The solutions are x = 3 and x = 2/3

4 3x² - 11x + 8 = 0

Hint: Identify the coefficients a, b, and c from the standard form ax² + bx + c = 0, then substitute them into the quadratic formula.

Show the answer

Answer: x = 2.667, x = 1

  1. Identify coefficients: a = 3, b = -11, c = 8
  2. Write the quadratic formula: x = [-b ± √(b² - 4ac)] / (2a)
  3. Substitute the values: x = [11 ± √((-11)² - 4*3*8)] / (2*3)
  4. Calculate the discriminant: (-11)² - 4*3*8 = 121 - 96 = 25
  5. Take the square root: √25 = 5
  6. Apply the formula: x = [11 ± 5] / 6
  7. Calculate the two solutions: x = (11 + 5) / 6 = 16 / 6 = 8/3 = 2.667 x = (11 - 5) / 6 = 6 / 6 = 1
  8. The solutions are x = 8/3 (or 2.667) and x = 1

5 3x² + 7x - 11 = 0

Hint: Identify coefficients a, b, and c from the standard form ax² + bx + c = 0, then substitute them into the quadratic formula x = [-b ± √(b² - 4ac)] / (2a).

Show the answer

Answer: x = (-7 + √157)/6, x = (-7 - √157)/6

  1. Identify coefficients: a = 3, b = 7, c = -11
  2. Calculate discriminant: b² - 4ac = 7² - 4(3)(-11) = 49 + 132 = 181
  3. Apply quadratic formula: x = [-7 ± √181] / (2×3)
  4. Simplify denominator: x = [-7 ± √181] / 6
  5. Write final solutions: x = (-7 + √181)/6 and x = (-7 - √181)/6

6 7x² - 11x + 4 = 0

Hint: For a quadratic equation ax² + bx + c = 0, use the quadratic formula x = [-b ± √(b² - 4ac)] / (2a). Identify the coefficients a, b, and c from your equation.

Show the answer

Answer: x = 1, x = 4/7

  1. Identify coefficients: a = 7, b = -11, c = 4
  2. Calculate discriminant: b² - 4ac = (-11)² - 4(7)(4) = 121 - 112 = 9
  3. Apply quadratic formula: x = [11 ± √9] / (2×7) = [11 ± 3] / 14
  4. Calculate both solutions: x = (11 + 3)/14 = 14/14 = 1, x = (11 - 3)/14 = 8/14 = 4/7
  5. Final solutions: x = 1 and x = 4/7
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