Solve (2x+1)(x+8)=27

$$( 2 x + 1 ) ( x + 8 ) = 27$$

Solve for x

$$x = -\frac{19}{2} = -9\frac{1}{2} = -9.5$$

$$x=1$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 2x+1 by x+8 and combine like terms.

$$2x^{2}+17x+8=27$$

Subtract 27 from both sides.

$$2x^{2}+17x+8-27=0$$

Subtract 27 from 8 to get -19.

$$2x^{2}+17x-19=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 17 for b, and -19 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-17±\sqrt{17^{2}-4\times 2\left(-19\right)}}{2\times 2}$$

Square 17.

$$x=\frac{-17±\sqrt{289-4\times 2\left(-19\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-17±\sqrt{289-8\left(-19\right)}}{2\times 2}$$

Multiply -8 times -19.

$$x=\frac{-17±\sqrt{289+152}}{2\times 2}$$

Add 289 to 152.

$$x=\frac{-17±\sqrt{441}}{2\times 2}$$

Take the square root of 441.

$$x=\frac{-17±21}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-17±21}{4}$$

Now solve the equation x=\frac{-17±21}{4} when ± is plus. Add -17 to 21.

$$x=\frac{4}{4}$$

Divide 4 by 4.

$$x=1$$

Now solve the equation x=\frac{-17±21}{4} when ± is minus. Subtract 21 from -17.

$$x=-\frac{38}{4}$$

Reduce the fraction \frac{-38}{4} to lowest terms by extracting and canceling out 2.

$$x=-\frac{19}{2}$$

The equation is now solved.

$$x=1 x=-\frac{19}{2}$$

Steps for Completing the Square

Use the distributive property to multiply 2x+1 by x+8 and combine like terms.

$$2x^{2}+17x+8=27$$

Subtract 8 from both sides.

$$2x^{2}+17x=27-8$$

Subtract 8 from 27 to get 19.

$$2x^{2}+17x=19$$

Divide both sides by 2.

$$\frac{2x^{2}+17x}{2}=\frac{19}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{17}{2}x=\frac{19}{2}$$

Divide \frac{17}{2}, the coefficient of the x term, by 2 to get \frac{17}{4}. Then add the square of \frac{17}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{17}{2}x+\left(\frac{17}{4}\right)^{2}=\frac{19}{2}+\left(\frac{17}{4}\right)^{2}$$

Square \frac{17}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{17}{2}x+\frac{289}{16}=\frac{19}{2}+\frac{289}{16}$$

Add \frac{19}{2} to \frac{289}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{17}{2}x+\frac{289}{16}=\frac{441}{16}$$

Factor x^{2}+\frac{17}{2}x+\frac{289}{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}.

$$\left(x+\frac{17}{4}\right)^{2}=\frac{441}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{17}{4}\right)^{2}}=\sqrt{\frac{441}{16}}$$

Simplify.

$$x+\frac{17}{4}=\frac{21}{4} x+\frac{17}{4}=-\frac{21}{4}$$

Subtract \frac{17}{4} from both sides of the equation.

$$x=1 x=-\frac{19}{2}$$