Solve x² - 5 = 0

Solve the quadratic equation x² - 5 = 0 using the quadratic formula.

Solution:

Discriminant: Δ = 20 (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 = 0 (coefficient of x)

• c = -5 (constant term)

Standard form: x² - 5 = 0

x² - 5 = 0

Step 2 Calculate the Discriminant

The discriminant determines the nature of the roots.

D = b² - 4ac

D = (0)² - 4(1)(-5)

D = 0 - -20

D = 20

Since D = 20 > 0, the equation has two distinct REAL roots.

D = 20

Step 3 Solve by Square Root Method

Since b = 0, this is a pure quadratic equation:

1x² + -5 = 0

We can solve directly by isolating x².

1x^2 + -5 = 0

Step 4 Isolate x² and Take Square Root

1x² = 5

x² = 5

Taking the square root of both sides:

x = ±√5

x₁ = +2.236068

x₂ = -2.236068

x = \pm2.236068

Step 5 Verify the Solutions

Substitute each solution back into the original equation:

For x = 2.236068:

1(2.236068)² + 0(2.236068) + -5

= 0 ✓

For x = -2.236068:

1(-2.236068)² + 0(-2.236068) + -5

= 0 ✓

\text{Both solutions verified!}

Step 6 Final Answer

x = -2.236068 or x = 2.236068