How to Use the Quadratic Formula

Step-by-step solution

1. Identify a, b and c.

   $$a = 2,\quad b = 3,\quad c = -2$$

   Why: The coefficients come from ax^2 + bx + c = 0.

2. Substitute into the discriminant.

   $$b^2 - 4ac = 3^2 - 4(2)(-2) = 9 + 16 = 25$$

   Why: The discriminant sits under the square root and tells you how many solutions exist.

3. Take the square root.

   $$\sqrt{25} = 5$$

   Why: A positive discriminant gives two real solutions.

4. Substitute into the quadratic formula.

   $$x = \frac{-3 \pm 5}{2(2)} = \frac{-3 \pm 5}{4}$$

   Why: The formula uses a, b and the square root of the discriminant.

5. Work out both values of x.

   $$x = \frac{2}{4} = \frac{1}{2},\quad x = \frac{-8}{4} = -2$$

   Why: The plus-or-minus sign gives the two solutions.

Practice problems

Use the same method. Work each problem on paper, then open the answer to check.

1. 1

   $x^2 + 3x + 2 = 0$

   Show answer

   $x = -1,\ x = -2$

2. 2

   $x^2 - 4x + 3 = 0$

   Show answer

   $x = 1,\ x = 3$

3. 3

   $2x^2 - 7x + 3 = 0$

   Show answer

   $x = 3,\ x = \frac{1}{2}$

4. 4

   $x^2 + 2x - 8 = 0$

   Show answer

   $x = 2,\ x = -4$

5. 5

   $x^2 - 6x + 9 = 0$

   Show answer

   $x = 3$