Solve x² + 7x + 12 = 0

Solve the quadratic equation x² + 7x + 12 = 0 using the quadratic formula.

Solution:

Discriminant: Δ = 1 (2 real roots)

Step-by-Step Solution

Step 1 Identify the Standard Form

A quadratic equation has the form: ax² + bx + c = 0

From the given equation:

• a = 1 (coefficient of x²)

• b = 7 (coefficient of x)

• c = 12 (constant term)

Standard form: x² + 7x + 12 = 0

x² + 7x + 12 = 0

Step 2 Calculate the Discriminant

The discriminant determines the nature of the roots.

D = b² - 4ac

D = (7)² - 4(1)(12)

D = 49 - 48

D = 1

Since D = 1 > 0 and √D = 1 (perfect square),

the equation has two distinct RATIONAL roots.

D = 1

Step 3 Solve by Factoring

For x² + 7x + 12 = 0, we need two numbers that:

• Multiply to give c = 12

• Add to give b = 7

The numbers are: 3 and 4

Check: 3 × 4 = 12 ✓

Check: 3 + 4 = 7 ✓

\text{Find factors of } 12 \text{ that sum to } 7

Step 4 Write the Factored Form

x² + 7x + 12 = (x + 3)(x + 4)

Setting each factor equal to zero:

(x + 3) = 0 → x = -3

(x + 4) = 0 → x = -4

(x + 3)(x + 4) = 0

Step 5 Verify the Solutions

Substitute each solution back into the original equation:

For x = -3:

1(-3)² + 7(-3) + 12

= 0 ✓

For x = -4:

1(-4)² + 7(-4) + 12

= 0 ✓

\text{Both solutions verified!}

Step 6 Final Answer

x = -4 or x = -3