Solving to Complete the Square when a = 1 Solutions

•     Completing the square    is used to solve quadratic equations, particularly those that cannot be factored.
• To solve using completing the square, follow these steps:
  • Calculate the term that will make the expression    on the left    a perfect square trinomial.
  • Take the square root of    both sides    of the equation.
• Following these steps exactly shows that you can solve for the x-intercepts of quadratic equations    in many ways   .
• Here are some important tips to remember when solving by completing a square:
  • When finding a square root, write the    plus/minus symbol (±)    before the square root symbol because you are solving. 
  • Perform the    same    operation on    both    sides of the equation so that equality is maintained. (You can see this especially in steps 2, 3, 4, and 7 in Example 3.)
  • Answers are the     x-intercepts    and are also called solutions or roots.
  • Answers can be    real, imaginary, or complex   .
• The steps are the same for    every    completing the square problem, but the expressions will be    different    depending on the values of a, b, and c.

Note

Some examples of expressions that can be different when completing a square problem include simplifying radicals, rationalizing denominators, and working with the imaginary unit i.

Example 3

Solve by completing the square.

$x^2–31=8x$

Implement	Explain (in Steps)
$x^2–8x–31=0$	Write equation in standard form
$a=1,$ continue to step 3	Divide all terms by leading coefficient, a (If $a=\mathrm{1,}$ this step can be skipped)
$x^2–8x=31$	Add $–\frac{c}{a}$ to both sides of equation
$x^2–8x+\boxed{ {\left(–4\right)}^2 }=31+\boxed{ {\left(–4\right)}^2 }$	Simplify and write $+\boxed{ }$ after terms on both sides Calculate value that makes LEFT side of equation a perfect square trinomial

Note

$+\boxed{ }={\left(\frac{–8}{2}\right)}^2={\left(–4\right)}^2=16$

This step can be completed using mental math. If work is shown, it is often done off to the side so that the rest of the problem flows together.

 

Remember to add the value ${\left(\frac{b}{2}\right)}^2$ to the blanks on both sides of the equation. 

$$\begin{gathered} {\left(x–4\right)}^2=31+16 \\ {\left(x–4\right)}^2=47 \end{gathered}$$	Write left side of the equation as product of a binomial squared: ${\left(x–\frac{b}{2}\right)}^2$ Simplify right side of equation
$\sqrt{{\left(x–4\right)}^2}=\pm \sqrt{47}$	Solve for x by taking square root of entire equation
$x–4=\pm \sqrt{47}$	Isolate the variable
$x=4\pm \sqrt{47}$

Example 4

Solve by completing the square.

$x^2+3x+2=–2x$

${\sqrt{\left(x+\frac{5}{2}\right)}}^2=\pm \sqrt{\frac{17}{4}}$

$x+\frac{5}{2}=\pm \frac{\sqrt{17}}{2}$

$x=–\frac{5}{2}\pm \frac{\sqrt{17}}{2}$