Solve x² - 2 = 0

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

Solution:

Discriminant: Δ = 8 (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 = -2 (constant term)

Standard form: x² - 2 = 0

x² - 2 = 0

Step 2 Calculate the Discriminant

The discriminant determines the nature of the roots.

D = b² - 4ac

D = (0)² - 4(1)(-2)

D = 0 - -8

D = 8

Since D = 8 > 0, the equation has two distinct REAL roots.

D = 8

Step 3 Solve by Square Root Method

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

1x² + -2 = 0

We can solve directly by isolating x².

1x^2 + -2 = 0

Step 4 Isolate x² and Take Square Root

1x² = 2

x² = 2

Taking the square root of both sides:

x = ±√2

x₁ = +1.414214

x₂ = -1.414214

x = \pm1.414214

Step 5 Verify the Solutions

Substitute each solution back into the original equation:

For x = 1.414214:

1(1.414214)² + 0(1.414214) + -2

= 0 ✓

For x = -1.414214:

1(-1.414214)² + 0(-1.414214) + -2

= 0 ✓

\text{Both solutions verified!}

Step 6 Final Answer

x = -1.414214 or x = 1.414214