Solve x² + 5x + 6 = 0

Solve the quadratic equation x² + 5x + 6 = 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 = 5 (coefficient of x)

• c = 6 (constant term)

Standard form: x² + 5x + 6 = 0

x² + 5x + 6 = 0

Step 2 Calculate the Discriminant

The discriminant determines the nature of the roots.

D = b² - 4ac

D = (5)² - 4(1)(6)

D = 25 - 24

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 + 5x + 6 = 0, we need two numbers that:

• Multiply to give c = 6

• Add to give b = 5

The numbers are: 2 and 3

Check: 2 × 3 = 6 ✓

Check: 2 + 3 = 5 ✓

\text{Find factors of } 6 \text{ that sum to } 5

Step 4 Write the Factored Form

x² + 5x + 6 = (x + 2)(x + 3)

Setting each factor equal to zero:

(x + 2) = 0 → x = -2

(x + 3) = 0 → x = -3

(x + 2)(x + 3) = 0

Step 5 Verify the Solutions

Substitute each solution back into the original equation:

For x = -2:

1(-2)² + 5(-2) + 6

= 0 ✓

For x = -3:

1(-3)² + 5(-3) + 6

= 0 ✓

\text{Both solutions verified!}

Step 6 Final Answer

x = -3 or x = -2