Solve x² - 16 = 0
Solve the quadratic equation x² - 16 = 0 using the quadratic formula.
Solution:
Discriminant: Δ = 64 (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 = -16 (constant term)
Standard form: x² - 16 = 0
x² - 16 = 0
Step 2
Calculate the Discriminant
The discriminant determines the nature of the roots.
D = b² - 4ac
D = (0)² - 4(1)(-16)
D = 0 - -64
D = 64
Since D = 64 > 0 and √D = 8 (perfect square),
the equation has two distinct RATIONAL roots.
D = 64
Step 3
Solve by Square Root Method
Since b = 0, this is a pure quadratic equation:
1x² + -16 = 0
We can solve directly by isolating x².
1x^2 + -16 = 0
Step 4
Isolate x² and Take Square Root
1x² = 16
x² = 16
Taking the square root of both sides:
x = ±√16
x₁ = +4
x₂ = -4
x = \pm4
Step 5
Verify the Solutions
Substitute each solution back into the original equation:
For x = 4:
1(4)² + 0(4) + -16
= 0 ✓
For x = -4:
1(-4)² + 0(-4) + -16
= 0 ✓
\text{Both solutions verified!}
Step 6
Final Answer
x = -4 or x = 4