Solve x² - 9 = 0
Solve the quadratic equation x² - 9 = 0 using the quadratic formula.
Solution:
Discriminant: Δ = 36 (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 = -9 (constant term)
Standard form: x² - 9 = 0
x² - 9 = 0
Step 2
Calculate the Discriminant
The discriminant determines the nature of the roots.
D = b² - 4ac
D = (0)² - 4(1)(-9)
D = 0 - -36
D = 36
Since D = 36 > 0 and √D = 6 (perfect square),
the equation has two distinct RATIONAL roots.
D = 36
Step 3
Solve by Square Root Method
Since b = 0, this is a pure quadratic equation:
1x² + -9 = 0
We can solve directly by isolating x².
1x^2 + -9 = 0
Step 4
Isolate x² and Take Square Root
1x² = 9
x² = 9
Taking the square root of both sides:
x = ±√9
x₁ = +3
x₂ = -3
x = \pm3
Step 5
Verify the Solutions
Substitute each solution back into the original equation:
For x = 3:
1(3)² + 0(3) + -9
= 0 ✓
For x = -3:
1(-3)² + 0(-3) + -9
= 0 ✓
\text{Both solutions verified!}
Step 6
Final Answer
x = -3 or x = 3