Solve x² - 1 = 0
Solve the quadratic equation x² - 1 = 0 using the quadratic formula.
Solution:
Discriminant: Δ = 4 (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 = -1 (constant term)
Standard form: x² - 1 = 0
x² - 1 = 0
Step 2
Calculate the Discriminant
The discriminant determines the nature of the roots.
D = b² - 4ac
D = (0)² - 4(1)(-1)
D = 0 - -4
D = 4
Since D = 4 > 0 and √D = 2 (perfect square),
the equation has two distinct RATIONAL roots.
D = 4
Step 3
Solve by Square Root Method
Since b = 0, this is a pure quadratic equation:
1x² + -1 = 0
We can solve directly by isolating x².
1x^2 + -1 = 0
Step 4
Isolate x² and Take Square Root
1x² = 1
x² = 1
Taking the square root of both sides:
x = ±√1
x₁ = +1
x₂ = -1
x = \pm1
Step 5
Verify the Solutions
Substitute each solution back into the original equation:
For x = 1:
1(1)² + 0(1) + -1
= 0 ✓
For x = -1:
1(-1)² + 0(-1) + -1
= 0 ✓
\text{Both solutions verified!}
Step 6
Final Answer
x = -1 or x = 1