Quadratic equations

Concept

A quadratic equation has the form ax² + bx + c = 0 where a ≠ 0. Three reliable solving methods:

Factoring — rewrite as (x − p)(x − q) = 0, giving roots p and q. Quickest when factors are obvious.
Quadratic formula — x = (−b ± √(b² − 4ac)) / (2a). Always works.
Completing the square — rewrite the left side as a perfect square (x − h)² = k, then take the square root.

The discriminant Δ = b² − 4ac tells you how many real solutions to expect: Δ > 0 two distinct, Δ = 0 one double root, Δ < 0 no real solutions.

Solution

Step 1. Find two numbers that multiply to 12 and sum to −7. Those are −3 and −4.

Step 2. Factor: (x − 3)(x − 4) = 0

Step 3. Set each factor to zero: x = 3 or x = 4.

x = 3 or x = 4. Check: 3² − 7(3) + 12 = 0 ✓ and 4² − 7(4) + 12 = 0 ✓

Solution

Identify. a = 2, b = −4, c = −3.

Discriminant. Δ = (−4)² − 4(2)(−3) = 16 + 24 = 40. Positive ⇒ two real roots.

Apply. x = (4 ± √40) / 4 = (4 ± 2√10) / 4 = (2 ± √10) / 2

x = (2 + √10)/2 ≈ 2.58 or x = (2 − √10)/2 ≈ −0.58.

8 questions on factoring, the quadratic formula, completing the square, and the discriminant.

Practice test · 8 questions Question 1 of 8 · Score 0