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H35 • Lesson 57 of 105

Solving Quadratics

Factoring, quadratic formula, completing the square

High School Essentials • 9-12

Prerequisites: H34

Key Concepts

  • solving
  • quadratic formula
  • methods

Solving Quadratics

Now that you understand what quadratic equations look like and how their graphs behave, it is time to learn how to solve them. Solving ax2 + bx + c = 0 means finding the values of x that make the equation true -- the roots, zeros, or x-intercepts. You have three primary tools: factoring, the quadratic formula, and completing the square.

Method 1: Solving by Factoring

If you can factor the quadratic, this is often the fastest method. It relies on the Zero Product Property: if AB = 0, then A = 0 or B = 0.

Worked Example 1 -- Solving by Factoring

Solve x2 - 5x + 6 = 0.

  1. Factor: find two numbers that multiply to 6 and add to -5. Those are -2 and -3.
  2. Write as (x - 2)(x - 3) = 0.
  3. Apply the Zero Product Property: x - 2 = 0 or x - 3 = 0.
  4. Solve: x = 2 or x = 3.

Check: 22 - 5(2) + 6 = 4 - 10 + 6 = 0. And 32 - 5(3) + 6 = 9 - 15 + 6 = 0. Both check out.

Method 2: The Quadratic Formula

The quadratic formula works for every quadratic equation, whether or not it factors neatly:

x = [-b ± sqrt(b2 - 4ac)] / (2a)

The expression under the square root, b2 - 4ac, is the discriminant, which tells you the nature of the solutions.

Worked Example 2 -- Quadratic Formula

Solve 2x2 + 3x - 5 = 0.

  1. Identify a = 2, b = 3, c = -5.
  2. Calculate the discriminant: D = 32 - 4(2)(-5) = 9 + 40 = 49.
  3. Apply the formula: x = [-3 ± sqrt(49)] / (2 * 2) = (-3 ± 7) / 4.
  4. Two solutions: x = (-3 + 7)/4 = 4/4 = 1 and x = (-3 - 7)/4 = -10/4 = -5/2.

Solutions: x = 1 or x = -5/2.

Method 3: Completing the Square

This method transforms ax2 + bx + c = 0 into the form (x + p)2 = q, making x easy to isolate.

  1. Move the constant to the right side.
  2. If a is not 1, divide every term by a.
  3. Take half of the x-coefficient, square it, and add it to both sides.
  4. Factor the left side as a perfect square.
  5. Take the square root of both sides (do not forget the ±).
  6. Solve for x.

Worked Example 3 -- Completing the Square

Solve x2 + 8x + 5 = 0.

  1. Move the constant: x2 + 8x = -5.
  2. Half of 8 is 4; 42 = 16. Add 16 to both sides: x2 + 8x + 16 = -5 + 16 = 11.
  3. Factor the left side: (x + 4)2 = 11.
  4. Take square roots: x + 4 = ±sqrt(11).
  5. Solve: x = -4 ± sqrt(11).

The exact solutions are x = -4 + sqrt(11) and x = -4 - sqrt(11), approximately x ≈ -0.68 and x ≈ -7.32.

Choosing the Best Method

Factoring: Use when the quadratic factors easily over the integers. Fastest when it works, but not all quadratics factor neatly.

Quadratic formula: The universal method. Always works. Use it when factoring is not obvious or when you need exact answers with radicals.

Completing the square: Most useful when you need vertex form or when b is even (making the arithmetic cleaner). Also needed to derive the quadratic formula itself.

Common Mistake: Forgetting the ±

When taking the square root of both sides, you must include both the positive and negative roots. Writing (x + 4)2 = 11 and then x + 4 = sqrt(11) gives you only one solution. The correct step is x + 4 = ±sqrt(11).

Practice Problems

1. Solve by factoring: x2 + x - 12 = 0.

Show Solution

Factor: (x + 4)(x - 3) = 0. Solutions: x = -4 or x = 3.

2. Solve using the quadratic formula: x2 - 6x + 2 = 0.

Show Solution

a = 1, b = -6, c = 2. D = 36 - 8 = 28. x = (6 ± sqrt(28)) / 2 = (6 ± 2sqrt(7)) / 2 = 3 ± sqrt(7). Approximately x ≈ 5.65 or x ≈ 0.35.

3. Solve by completing the square: x2 - 10x + 21 = 0.

Show Solution

x2 - 10x = -21. Half of -10 is -5; (-5)2 = 25. Add 25: x2 - 10x + 25 = 4. Factor: (x - 5)2 = 4. Take roots: x - 5 = ±2. So x = 7 or x = 3.

4. Solve: 3x2 + 6x = 0.

Show Solution

Factor out 3x: 3x(x + 2) = 0. By the Zero Product Property: 3x = 0 or x + 2 = 0. So x = 0 or x = -2.

5. How many real solutions does 4x2 - 4x + 1 = 0 have? Solve it.

Show Solution

D = (-4)2 - 4(4)(1) = 16 - 16 = 0. One repeated real solution. Using the formula: x = 4/(2*4) = 4/8 = 1/2. Or factor: (2x - 1)2 = 0, so x = 1/2.

Summary

Three methods solve quadratic equations: factoring (fastest when possible), the quadratic formula (always works), and completing the square (useful for exact form and deriving vertex form). The discriminant b2 - 4ac predicts the number and type of solutions. Always check your solutions by substituting them back into the original equation, and never forget the ± when taking square roots.

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