Solve (x)(x+9)=19

$$(x)(x+9)=19$$

Solve for x

$$x = \frac{\sqrt{157} - 9}{2} \approx 1.764982043$$

$$x=\frac{-\sqrt{157}-9}{2}\approx -10.764982043$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x by x+9.

$$x^{2}+9x=19$$

Subtract 19 from both sides.

$$x^{2}+9x-19=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 9 for b, and -19 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-9±\sqrt{9^{2}-4\left(-19\right)}}{2}$$

Square 9.

$$x=\frac{-9±\sqrt{81-4\left(-19\right)}}{2}$$

Multiply -4 times -19.

$$x=\frac{-9±\sqrt{81+76}}{2}$$

Add 81 to 76.

$$x=\frac{-9±\sqrt{157}}{2}$$

Now solve the equation x=\frac{-9±\sqrt{157}}{2} when ± is plus. Add -9 to \sqrt{157}.

$$x=\frac{\sqrt{157}-9}{2}$$

Now solve the equation x=\frac{-9±\sqrt{157}}{2} when ± is minus. Subtract \sqrt{157} from -9.

$$x=\frac{-\sqrt{157}-9}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{157}-9}{2} x=\frac{-\sqrt{157}-9}{2}$$

Steps for Completing the Square

Use the distributive property to multiply x by x+9.

$$x^{2}+9x=19$$

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.

$$x^{2}+9x+\left(\frac{9}{2}\right)^{2}=19+\left(\frac{9}{2}\right)^{2}$$

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

$$x^{2}+9x+\frac{81}{4}=19+\frac{81}{4}$$

Add 19 to \frac{81}{4}.

$$x^{2}+9x+\frac{81}{4}=\frac{157}{4}$$

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}.

$$\left(x+\frac{9}{2}\right)^{2}=\frac{157}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{9}{2}\right)^{2}}=\sqrt{\frac{157}{4}}$$

Simplify.

$$x+\frac{9}{2}=\frac{\sqrt{157}}{2} x+\frac{9}{2}=-\frac{\sqrt{157}}{2}$$

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

$$x=\frac{\sqrt{157}-9}{2} x=\frac{-\sqrt{157}-9}{2}$$