Solve (x+3)^2+5x=8

$$( x + 3 ) ^ { 2 } + 5 x = 8$$

Solve for x

$$x=\frac{3\sqrt{13}-11}{2}\approx -0.091673087$$

$$x=\frac{-3\sqrt{13}-11}{2}\approx -10.908326913$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+3\right)^{2}.

$$x^{2}+6x+9+5x=8$$

Combine 6x and 5x to get 11x.

$$x^{2}+11x+9=8$$

Subtract 8 from both sides.

$$x^{2}+11x+9-8=0$$

Subtract 8 from 9 to get 1.

$$x^{2}+11x+1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 11 for b, and 1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-11±\sqrt{11^{2}-4}}{2}$$

Square 11.

$$x=\frac{-11±\sqrt{121-4}}{2}$$

Add 121 to -4.

$$x=\frac{-11±\sqrt{117}}{2}$$

Take the square root of 117.

$$x=\frac{-11±3\sqrt{13}}{2}$$

Now solve the equation x=\frac{-11±3\sqrt{13}}{2} when ± is plus. Add -11 to 3\sqrt{13}.

$$x=\frac{3\sqrt{13}-11}{2}$$

Now solve the equation x=\frac{-11±3\sqrt{13}}{2} when ± is minus. Subtract 3\sqrt{13} from -11.

$$x=\frac{-3\sqrt{13}-11}{2}$$

The equation is now solved.

$$x=\frac{3\sqrt{13}-11}{2} x=\frac{-3\sqrt{13}-11}{2}$$

Steps for Completing the Square

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+3\right)^{2}.

$$x^{2}+6x+9+5x=8$$

Combine 6x and 5x to get 11x.

$$x^{2}+11x+9=8$$

Subtract 9 from both sides.

$$x^{2}+11x=8-9$$

Subtract 9 from 8 to get -1.

$$x^{2}+11x=-1$$

Divide 11, the coefficient of the x term, by 2 to get \frac{11}{2}. Then add the square of \frac{11}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+11x+\left(\frac{11}{2}\right)^{2}=-1+\left(\frac{11}{2}\right)^{2}$$

Square \frac{11}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+11x+\frac{121}{4}=-1+\frac{121}{4}$$

Add -1 to \frac{121}{4}.

$$x^{2}+11x+\frac{121}{4}=\frac{117}{4}$$

Factor x^{2}+11x+\frac{121}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{11}{2}\right)^{2}=\frac{117}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{11}{2}\right)^{2}}=\sqrt{\frac{117}{4}}$$

Simplify.

$$x+\frac{11}{2}=\frac{3\sqrt{13}}{2} x+\frac{11}{2}=-\frac{3\sqrt{13}}{2}$$

Subtract \frac{11}{2} from both sides of the equation.

$$x=\frac{3\sqrt{13}-11}{2} x=\frac{-3\sqrt{13}-11}{2}$$