Solve (x+3)(x+6)=16

$$( x + 3 ) ( x + 6 ) = 16$$

Solve for x

$$x=\frac{\sqrt{73}-9}{2}\approx -0.227998127$$

$$x=\frac{-\sqrt{73}-9}{2}\approx -8.772001873$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x+3 by x+6 and combine like terms.

$$x^{2}+9x+18=16$$

Subtract 16 from both sides.

$$x^{2}+9x+18-16=0$$

Subtract 16 from 18 to get 2.

$$x^{2}+9x+2=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 9 for b, and 2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-9±\sqrt{9^{2}-4\times 2}}{2}$$

Square 9.

$$x=\frac{-9±\sqrt{81-4\times 2}}{2}$$

Multiply -4 times 2.

$$x=\frac{-9±\sqrt{81-8}}{2}$$

Add 81 to -8.

$$x=\frac{-9±\sqrt{73}}{2}$$

Now solve the equation x=\frac{-9±\sqrt{73}}{2} when ± is plus. Add -9 to \sqrt{73}.

$$x=\frac{\sqrt{73}-9}{2}$$

Now solve the equation x=\frac{-9±\sqrt{73}}{2} when ± is minus. Subtract \sqrt{73} from -9.

$$x=\frac{-\sqrt{73}-9}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{73}-9}{2} x=\frac{-\sqrt{73}-9}{2}$$

Steps for Completing the Square

Use the distributive property to multiply x+3 by x+6 and combine like terms.

$$x^{2}+9x+18=16$$

Subtract 18 from both sides.

$$x^{2}+9x=16-18$$

Subtract 18 from 16 to get -2.

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

Divide 9, the coefficient of the x term, by 2 to get \frac{9}{2}. Then add the square of \frac{9}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+9x+\left(\frac{9}{2}\right)^{2}=-2+\left(\frac{9}{2}\right)^{2}$$

Square \frac{9}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+9x+\frac{81}{4}=-2+\frac{81}{4}$$

Add -2 to \frac{81}{4}.

$$x^{2}+9x+\frac{81}{4}=\frac{73}{4}$$

Factor x^{2}+9x+\frac{81}{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}.

$$\left(x+\frac{9}{2}\right)^{2}=\frac{73}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{9}{2}\right)^{2}}=\sqrt{\frac{73}{4}}$$

Simplify.

$$x+\frac{9}{2}=\frac{\sqrt{73}}{2} x+\frac{9}{2}=-\frac{\sqrt{73}}{2}$$

Subtract \frac{9}{2} from both sides of the equation.

$$x=\frac{\sqrt{73}-9}{2} x=\frac{-\sqrt{73}-9}{2}$$