Solve (2x+1)(x+5)=-3

$$( 2 x + 1 ) ( x + 5 ) = - 3$$

Solve for x

$$x=\frac{\sqrt{57}-11}{4}\approx -0.862541391$$

$$x=\frac{-\sqrt{57}-11}{4}\approx -4.637458609$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 2x+1 by x+5 and combine like terms.

$$2x^{2}+11x+5=-3$$

Add 3 to both sides.

$$2x^{2}+11x+5+3=0$$

Add 5 and 3 to get 8.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 11 for b, and 8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-11±\sqrt{11^{2}-4\times 2\times 8}}{2\times 2}$$

Square 11.

$$x=\frac{-11±\sqrt{121-4\times 2\times 8}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-11±\sqrt{121-8\times 8}}{2\times 2}$$

Multiply -8 times 8.

$$x=\frac{-11±\sqrt{121-64}}{2\times 2}$$

Add 121 to -64.

$$x=\frac{-11±\sqrt{57}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-11±\sqrt{57}}{4}$$

Now solve the equation x=\frac{-11±\sqrt{57}}{4} when ± is plus. Add -11 to \sqrt{57}.

$$x=\frac{\sqrt{57}-11}{4}$$

Now solve the equation x=\frac{-11±\sqrt{57}}{4} when ± is minus. Subtract \sqrt{57} from -11.

$$x=\frac{-\sqrt{57}-11}{4}$$

The equation is now solved.

$$x=\frac{\sqrt{57}-11}{4} x=\frac{-\sqrt{57}-11}{4}$$

Steps for Completing the Square

Use the distributive property to multiply 2x+1 by x+5 and combine like terms.

$$2x^{2}+11x+5=-3$$

Subtract 5 from both sides.

$$2x^{2}+11x=-3-5$$

Subtract 5 from -3 to get -8.

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

Divide both sides by 2.

$$\frac{2x^{2}+11x}{2}=-\frac{8}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{11}{2}x=-\frac{8}{2}$$

Divide -8 by 2.

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

Divide \frac{11}{2}, the coefficient of the x term, by 2 to get \frac{11}{4}. Then add the square of \frac{11}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square \frac{11}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{11}{2}x+\frac{121}{16}=-4+\frac{121}{16}$$

Add -4 to \frac{121}{16}.

$$x^{2}+\frac{11}{2}x+\frac{121}{16}=\frac{57}{16}$$

Factor x^{2}+\frac{11}{2}x+\frac{121}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{11}{4}\right)^{2}=\frac{57}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{11}{4}\right)^{2}}=\sqrt{\frac{57}{16}}$$

Simplify.

$$x+\frac{11}{4}=\frac{\sqrt{57}}{4} x+\frac{11}{4}=-\frac{\sqrt{57}}{4}$$

Subtract \frac{11}{4} from both sides of the equation.

$$x=\frac{\sqrt{57}-11}{4} x=\frac{-\sqrt{57}-11}{4}$$