Math Problem Statement

complete the swuare

Solve x^2 - 4x - 13 = 0 by completing the square.

Solution

To solve x24x13=0x^2 - 4x - 13 = 0 by completing the square, follow these steps:

Step 1: Rearrange the equation

Move the constant term (13-13) to the other side of the equation: x24x=13x^2 - 4x = 13

Step 2: Complete the square

Take half of the coefficient of xx, square it, and add it to both sides. The coefficient of xx is 4-4, so: (42)2=4\left(\frac{-4}{2}\right)^2 = 4 Add 44 to both sides: x24x+4=13+4x^2 - 4x + 4 = 13 + 4 (x2)2=17(x - 2)^2 = 17

Step 3: Solve for xx

Take the square root of both sides: x2=±17x - 2 = \pm \sqrt{17}

Step 4: Isolate xx

Add 22 to both sides: x=2±17x = 2 \pm \sqrt{17}

Final Answer:

The solutions are: x=2+17,x=217x = 2 + \sqrt{17}, \quad x = 2 - \sqrt{17}

If you have any further questions, let me know!


Related Questions:

  1. What does "completing the square" help us understand about quadratic equations?
  2. How can we verify that the solutions satisfy the original equation?
  3. What are some real-world applications of solving quadratic equations like this?
  4. How does completing the square relate to the quadratic formula?
  5. Could the solutions be expressed in decimal form? What would they approximately be?

Tip:

Always double-check your calculations when completing the square, especially when determining the term to add!

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Math Problem Analysis

Mathematical Concepts

Algebra
Completing the Square
Quadratic Equations

Formulas

(x + p)^2 = q
Quadratic solutions: x = h ± √k

Theorems

The Square Root Property

Suitable Grade Level

Grades 8-10