Solve (2x+3)(x-6)=16

$$( 2 x + 3 ) ( x - 6 ) = 16$$

Solve for x

$$x = \frac{\sqrt{353} + 9}{4} \approx 6.947073557$$

$$x=\frac{9-\sqrt{353}}{4}\approx -2.447073557$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 2x+3 by x-6 and combine like terms.

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

Subtract 16 from both sides.

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

Subtract 16 from -18 to get -34.

$$2x^{2}-9x-34=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -9 for b, and -34 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\times 2\left(-34\right)}}{2\times 2}$$

Square -9.

$$x=\frac{-\left(-9\right)±\sqrt{81-4\times 2\left(-34\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-9\right)±\sqrt{81-8\left(-34\right)}}{2\times 2}$$

Multiply -8 times -34.

$$x=\frac{-\left(-9\right)±\sqrt{81+272}}{2\times 2}$$

Add 81 to 272.

$$x=\frac{-\left(-9\right)±\sqrt{353}}{2\times 2}$$

The opposite of -9 is 9.

$$x=\frac{9±\sqrt{353}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{9±\sqrt{353}}{4}$$

Now solve the equation x=\frac{9±\sqrt{353}}{4} when ± is plus. Add 9 to \sqrt{353}.

$$x=\frac{\sqrt{353}+9}{4}$$

Now solve the equation x=\frac{9±\sqrt{353}}{4} when ± is minus. Subtract \sqrt{353} from 9.

$$x=\frac{9-\sqrt{353}}{4}$$

The equation is now solved.

$$x=\frac{\sqrt{353}+9}{4} x=\frac{9-\sqrt{353}}{4}$$

Steps for Completing the Square

Use the distributive property to multiply 2x+3 by x-6 and combine like terms.

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

Add 18 to both sides.

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

Add 16 and 18 to get 34.

$$2x^{2}-9x=34$$

Divide both sides by 2.

$$\frac{2x^{2}-9x}{2}=\frac{34}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}-\frac{9}{2}x=\frac{34}{2}$$

Divide 34 by 2.

$$x^{2}-\frac{9}{2}x=17$$

Divide -\frac{9}{2}, the coefficient of the x term, by 2 to get -\frac{9}{4}. Then add the square of -\frac{9}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{9}{2}x+\left(-\frac{9}{4}\right)^{2}=17+\left(-\frac{9}{4}\right)^{2}$$

Square -\frac{9}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{9}{2}x+\frac{81}{16}=17+\frac{81}{16}$$

Add 17 to \frac{81}{16}.

$$x^{2}-\frac{9}{2}x+\frac{81}{16}=\frac{353}{16}$$

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

$$\left(x-\frac{9}{4}\right)^{2}=\frac{353}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{9}{4}\right)^{2}}=\sqrt{\frac{353}{16}}$$

Simplify.

$$x-\frac{9}{4}=\frac{\sqrt{353}}{4} x-\frac{9}{4}=-\frac{\sqrt{353}}{4}$$

Add \frac{9}{4} to both sides of the equation.

$$x=\frac{\sqrt{353}+9}{4} x=\frac{9-\sqrt{353}}{4}$$