Math Problem Statement

Solve the equation by completing the square. x^2 - 8x + 16 = 0

Solution

Let's solve the quadratic equation x28x+16=0x^2 - 8x + 16 = 0 by completing the square.

Step 1: Write the equation in the form x28x=16x^2 - 8x = -16.

We already have: x28x+16=0x^2 - 8x + 16 = 0 Subtract 16 from both sides: x28x=16x^2 - 8x = -16

Step 2: Complete the square.

To complete the square, take half of the coefficient of xx, which is 8-8, divide by 2, and square it.

(82)2=16\left(\frac{-8}{2}\right)^2 = 16

Add 16 to both sides to complete the square:

x28x+16=0x^2 - 8x + 16 = 0

Now the left side is a perfect square trinomial:

(x4)2=0(x - 4)^2 = 0

Step 3: Solve for xx.

Take the square root of both sides:

x4=0x - 4 = 0

Solve for xx:

x=4x = 4

Thus, the solution is: x=4x = 4

Would you like more details on any step or have any questions?

Related Questions:

  1. How do you complete the square for a general quadratic equation?
  2. What if the quadratic equation had a different constant term, like x28x+12=0x^2 - 8x + 12 = 0?
  3. Can you apply completing the square for equations where the coefficient of x2x^2 is not 1?
  4. How does completing the square relate to the quadratic formula?
  5. What are other methods for solving quadratic equations?

Tip:

When completing the square, always ensure the coefficient of x2x^2 is 1. If it's not, divide the whole equation by the coefficient before proceeding!

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Math Problem Analysis

Mathematical Concepts

Algebra
Completing the Square
Quadratic Equations

Formulas

(x - b/2)^2 = (b/2)^2 - c

Theorems

The method of completing the square to solve quadratic equations

Suitable Grade Level

Grades 8-10