Lesson 2-4: The Quadratic Formula

Quadratic Formula

For any equation $ax² + bx + c = 0$ :

x = (-b \pm \sqrt(b² - 4ac)) / (2a)

This formula always gives the correct solutions — whether the quadratic factors nicely or not.

Solve $2x² - 5x + 2 = 0$ .

1. a = 2, b = -5, c = 2 2. x = (-(-5) \pm \sqrt((-5)² - 4(2)(2))) / (2\cdot2) 3. = (5 \pm \sqrt(25 - 16)) / 4 4. = (5 \pm \sqrt9) / 4 5. = (5 \pm 3) / 4 6. x = (5+3)/4 = 2 or x = (5-3)/4 = 1/2

The expression $b² - 4ac$ is called the discriminant (often written Δ). It tells you the number of real solutions before you fully solve.

Δ > 0

Two distinct real solutions

Δ = 0

One repeated real solution

Δ < 0

No real solutions

Example: For $x² + 2x + 5 = 0$ : Δ = 4 − 20 = −16 < 0 → no real solutions.

Example: For $x² - 6x + 9 = 0$ : Δ = 36 − 36 = 0 → one solution: x = 3.

Q1. Solve $x² + 4x - 5 = 0$ using the quadratic formula.

Show Answer

a=1, b=4, c=-5 Δ = 16 + 20 = 36 x = (-4 \pm 6) / 2 x = 1 or x = -5

Q2. Without fully solving, how many real solutions does $3x² - x + 2 = 0$ have?

Show Answer

Δ = (−1)² − 4(3)(2) = 1 − 24 = −23 < 0. No real solutions.