Solve x² - 25 = 0
Solve the quadratic equation x² - 25 = 0 using the quadratic formula.
Solution:
Discriminant: Δ = 100 (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 = -25 (constant term)
Standard form: x² - 25 = 0
x² - 25 = 0
Step 2
Calculate the Discriminant
The discriminant determines the nature of the roots.
D = b² - 4ac
D = (0)² - 4(1)(-25)
D = 0 - -100
D = 100
Since D = 100 > 0 and √D = 10 (perfect square),
the equation has two distinct RATIONAL roots.
D = 100
Step 3
Solve by Square Root Method
Since b = 0, this is a pure quadratic equation:
1x² + -25 = 0
We can solve directly by isolating x².
1x^2 + -25 = 0
Step 4
Isolate x² and Take Square Root
1x² = 25
x² = 25
Taking the square root of both sides:
x = ±√25
x₁ = +5
x₂ = -5
x = \pm5
Step 5
Verify the Solutions
Substitute each solution back into the original equation:
For x = 5:
1(5)² + 0(5) + -25
= 0 ✓
For x = -5:
1(-5)² + 0(-5) + -25
= 0 ✓
\text{Both solutions verified!}
Step 6
Final Answer
x = -5 or x = 5