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Chapter 0: Problem 71

Solve each quadratic equation by completing the square. $$x^{2}-6 x-11=0$$

Short Answer

Expert verified
The solution of the quadratic equation \(x^{2}-6x-11=0\) by completing the square is \(x=\sqrt{20}+3\) and \(x=-\sqrt{20}+3\).

Step by step solution

01

Rearrange the Equation

Rearrange the given equation \(x^{2}-6x-11=0\) such that the constant (in this case -11) is on the other side of the equation. This leads to \(x^{2}-6x=11\)
02

Completing the Square

The next part involves completing the square. Take half of the coefficient of x (i.e. \(-6/2 = -3\)), and square it (which gives 9). This needs to be added on both sides of the equation, leading to new equation \(x^{2}-6x+9=20\)
03

Solving the Squared Equation

The left side can now be rewritten as a binomial square and the square root can be applied on both sides to determine the value of x. This leads to \((x-3)^{2}=20\). Taking square roots on both sides will produce two roots as x can be either positive or negative. This leads to \(x-3=\sqrt{20}\) or \(x-3=-\sqrt{20}\). Solving these gives us two possible solutions for x.
04

Determining Final Results

Solving \(x-3=\sqrt{20}\), we have \(x=\sqrt{20}+3\), and solving \(x-3=-\sqrt{20}\), we get \(x=-\sqrt{20}+3\).

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