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Chapter 1: Problem 38

Solve each equation by completing the square $$ x^{2}-6 x=13 $$

Short Answer

Expert verified
The solutions are \( x = 3 + \sqrt{22} \) and \( x = 3 - \sqrt{22} \).

Step by step solution

01

Move constant term to the right side

First, rewrite the equation to isolate the terms involving x on one side. The given equation is \[ x^2 - 6x = 13 \] The right side already contains the constant term.
02

Add a constant to both sides to complete the square

To complete the square, take half of the coefficient of x, which is -6. Half of -6 is -3, and squaring it gives 9. Add 9 to both sides of the equation: \[ x^{2} - 6x + 9 = 13 + 9 \] This simplifies to: \[ x^2 - 6x + 9 = 22 \]
03

Express the left side as a perfect square

The left side of the equation can now be written as a squared binomial: \[ (x - 3)^2 = 22 \]
04

Solve for x

Take the square root of both sides of the equation: \[ x - 3 = \pm \sqrt{22} \] This gives two solutions: \[ x - 3 = \sqrt{22} \] and \[ x - 3 = -\sqrt{22} \] Solving for x in both cases: \[ x = 3 + \sqrt{22} \] and \[ x = 3 - \sqrt{22} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

solving quadratic equations
Solving quadratic equations involves finding the values of the variable (typically \(x\)) that make the equation true. Quadratic equations are often written in the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. Sometimes, the equation might not look like this right away, but with a few steps, it can be transformed into this standard form.

There are several methods to solve quadratic equations including:
  • Factoring
  • Using the quadratic formula
  • Completing the square
Each method has its own advantages depending on the structure of the quadratic equation. For example, completing the square can be very useful when the equation does not factor easily.
perfect square trinomial
A perfect square trinomial is a special type of quadratic equation where the quadratic expression can be written as the square of a binomial. For example, \(x^2 - 6x + 9\) can be rewritten as \((x - 3)^2\).

To recognize a perfect square trinomial, observe the form \(a^2 - 2ab + b^2\), which can be rewritten as \((a - b)^2\). In our example, comparing \(x^2 - 6x + 9\):
  • \(a^2 = x^2\) so \(a = x\)
  • \(b^2 = 9\) so \(b = 3\)
  • \(-2ab = -6x\)
When you add a specific constant term to complete the square, you're creating a perfect square trinomial. This step is key in the method of solving quadratic equations by completing the square.
quadratic formula
The quadratic formula provides a way to solve any quadratic equation and is very helpful when the equation is difficult to factor. The quadratic formula is:

\[ x = \frac{-b\pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \(a\), \(b\), and \(c\) are the coefficients from the equation \(ax^2 + bx + c = 0\). The term under the square root \(b^2 - 4ac\) is called the discriminant. It determines the nature of the roots (solutions) of the quadratic equation:
  • If the discriminant is positive, there are two real and distinct solutions.
  • If the discriminant is zero, there is exactly one real solution.
  • If the discriminant is negative, the equation has two complex solutions.
While the quadratic formula is powerful, understanding and completing the square method provides additional insight and reinforces understanding of quadratic relationships.

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