Solve x² - 1x - 6 = 0

Solve the quadratic equation x² - 1x - 6 = 0 using the quadratic formula.

Solution:

Discriminant: Δ = 25 (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 = -1 (coefficient of x)

• c = -6 (constant term)

Standard form: x² - x - 6 = 0

x² - x - 6 = 0

Step 2 Calculate the Discriminant

The discriminant determines the nature of the roots.

D = b² - 4ac

D = (-1)² - 4(1)(-6)

D = 1 - -24

D = 25

Since D = 25 > 0 and √D = 5 (perfect square),

the equation has two distinct RATIONAL roots.

D = 25

Step 3 Solve by Factoring

For x² + -1x + -6 = 0, we need two numbers that:

• Multiply to give c = -6

• Add to give b = -1

The numbers are: -3 and 2

Check: -3 × 2 = -6 ✓

Check: -3 + 2 = -1 ✓

\text{Find factors of } -6 \text{ that sum to } -1

Step 4 Write the Factored Form

x² + -1x + -6 = (x - 3)(x + 2)

Setting each factor equal to zero:

(x - 3) = 0 → x = 3

(x + 2) = 0 → x = -2

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

Step 5 Verify the Solutions

Substitute each solution back into the original equation:

For x = 3:

1(3)² + -1(3) + -6

= 0 ✓

For x = -2:

1(-2)² + -1(-2) + -6

= 0 ✓

\text{Both solutions verified!}

Step 6 Final Answer

x = -2 or x = 3