Q12. - Algebra 1

Question

Solve each equation by completing the square.

$x^2+2x+5=0$

Step-by-Step Solution

Answer

The quadratic equations have no real solutions.

Step 1. Complete the square.

Complete the square to solve the quadratic equations by isolating $x^2$ and b-terms.

$$\begin{gathered} x^2+2x+5=0 \\ x^2+2x+5+1=0+1 \left[\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}Since\hspace{0.17em}\hspace{0.17em}}{\left(\frac{2}{2}\right)}^{\text{2}}=1\text{\hspace{0.17em}\hspace{0.17em}add\hspace{0.17em}\hspace{0.17em}to\hspace{0.17em}\hspace{0.17em}each\hspace{0.17em}\hspace{0.17em}side}\right] \\ {x}^2+2x+1+5-5=1-5 \left[\text{\hspace{0.17em}\hspace{0.17em}Subtract\hspace{0.17em}\hspace{0.17em}5\hspace{0.17em}\hspace{0.17em}to\hspace{0.17em}\hspace{0.17em}each\hspace{0.17em}\hspace{0.17em}side\hspace{0.17em}\hspace{0.17em}}\right] \\ x^2+2x+1=-4 \end{gathered}$$

Step 2. Take the square root of each side.

The first factor $x^2+2x+1$, then take the square root of each side.

width="400" height="54" style="max-width: none; vertical-align: -28px;" $$\begin{gathered} {(x+1)}^2=-4 \\ \sqrt{{(x+1)}^2}=\pm \sqrt{-4} \left[\text{\hspace{0.17em}\hspace{0.17em}Take\hspace{0.17em}\hspace{0.17em}the\hspace{0.17em}\hspace{0.17em}square\hspace{0.17em}\hspace{0.17em}root\hspace{0.17em}\hspace{0.17em}of\hspace{0.17em}\hspace{0.17em}each\hspace{0.17em}\hspace{0.17em}side\hspace{0.17em}}\right] \end{gathered}$$

Step 3. Write the solutions.

No real number has a negative square. So, this equation $x^2+2x+5=0$ has no real solutions.