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The Quadratic Formula and Discriminant

A free Algebra I lesson from the “Quadratic Foundations” unit, with a worked example and practice problems including step-by-step solutions.

For any equation ax^2 + bx + c = 0 with a not equal to 0, the solutions are x = (-b plus or minus sqrt(b^2 - 4ac)) / (2a). The expression b^2 - 4ac is called the discriminant. If positive, there are two real solutions; if zero, exactly one; if negative, no real solutions.

What you'll learn

Why it matters: Projectile motion (when does a ball hit the ground?), break-even analysis (when does cost equal revenue?), and any quadratic word problem that does not factor cleanly all rely on the formula.

Worked example

Problem. Solve x^2 + 4x - 5 = 0 using the quadratic formula.

  1. Identify a = 1, b = 4, c = -5.
  2. Discriminant: 4^2 - 4(1)(-5) = 16 + 20 = 36.
  3. x = (-4 plus or minus sqrt(36)) / 2 = (-4 plus or minus 6) / 2.
  4. x = 1 or x = -5.

Answer: x = 1 or x = -5

Practice problems

1. Solve x^2 - 5x + 6 = 0 by the quadratic formula. Enter the larger solution.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Solve x^2 - 5x + 6 = 0 by the quadratic formula. Enter the larger solution.
  2. For quadratics, track the zeros, vertex, or coefficients so the algebra matches the graph feature being asked about.
  3. a=1, b=-5, c=6; discriminant 25 - 24 = 1.
  4. x = (5 plus or minus 1)/2 = 3 or 2.
  5. The larger is 3.
  6. Check the result by substituting or estimating: the response should match 3 and make sense in the original problem.

Answer: 3

2. Solve x^2 + 6x + 8 = 0. Enter the larger solution.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Solve x^2 + 6x + 8 = 0. Enter the larger solution.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. a=1, b=6, c=8; discriminant 36 - 32 = 4.
  4. x = (-6 plus or minus 2)/2 = -2 or -4.
  5. The larger is -2.
  6. Check the result by substituting or estimating: the response should match -2 and make sense in the original problem.

Answer: -2

3. Solve x^2 - 4x + 3 = 0. Enter the smaller solution.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Solve x^2 - 4x + 3 = 0. Enter the smaller solution.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Discriminant 16 - 12 = 4.
  4. x = (4 plus or minus 2)/2 = 3 or 1.
  5. The smaller is 1.
  6. Check the result by substituting or estimating: the response should match 1 and make sense in the original problem.

Answer: 1

4. Solve 2x^2 - 5x + 2 = 0. Enter the smaller solution as a fraction.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Solve 2x^2 - 5x + 2 = 0. Enter the smaller solution as a fraction.
  2. For fractions, use equivalent forms, common denominators, or reciprocals depending on the operation being used.
  3. a=2, b=-5, c=2; discriminant 25 - 16 = 9.
  4. x = (5 plus or minus 3)/4 = 2 or 1/2.
  5. The smaller is 1/2.
  6. Check the result by substituting or estimating: the response should match 1/2 and make sense in the original problem.

Answer: 1/2

5. Solve x^2 - 7x + 12 = 0. Enter the larger solution.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Solve x^2 - 7x + 12 = 0. Enter the larger solution.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Discriminant 49 - 48 = 1.
  4. x = (7 plus or minus 1)/2 = 4 or 3.
  5. The larger is 4.
  6. Check the result by substituting or estimating: the response should match 4 and make sense in the original problem.

Answer: 4

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