Solve x² + 2x - 3 = 0

Solve the quadratic equation x² + 2x - 3 = 0 using the quadratic formula.

Solution:

Discriminant: Δ = 16 (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 = 2 (coefficient of x)

• c = -3 (constant term)

Standard form: x² + 2x - 3 = 0

x² + 2x - 3 = 0

Step 2 Calculate the Discriminant

The discriminant determines the nature of the roots.

D = b² - 4ac

D = (2)² - 4(1)(-3)

D = 4 - -12

D = 16

Since D = 16 > 0 and √D = 4 (perfect square),

the equation has two distinct RATIONAL roots.

D = 16

Step 3 Solve by Factoring

For x² + 2x + -3 = 0, we need two numbers that:

• Multiply to give c = -3

• Add to give b = 2

The numbers are: -1 and 3

Check: -1 × 3 = -3 ✓

Check: -1 + 3 = 2 ✓

\text{Find factors of } -3 \text{ that sum to } 2

Step 4 Write the Factored Form

x² + 2x + -3 = (x - 1)(x + 3)

Setting each factor equal to zero:

(x - 1) = 0 → x = 1

(x + 3) = 0 → x = -3

(x - 1)(x + 3) = 0

Step 5 Verify the Solutions

Substitute each solution back into the original equation:

For x = 1:

1(1)² + 2(1) + -3

= 0 ✓

For x = -3:

1(-3)² + 2(-3) + -3

= 0 ✓

\text{Both solutions verified!}

Step 6 Final Answer

x = -3 or x = 1