Solve (8x-15)(12x+5)=-91

$$(8x-15)(12x+5)=-91$$

Solve for x

$$x=\frac{1}{8}=0.125$$

$$x = \frac{4}{3} = 1\frac{1}{3} \approx 1.333333333$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 8x-15 by 12x+5 and combine like terms.

$$96x^{2}-140x-75=-91$$

Add 91 to both sides.

$$96x^{2}-140x-75+91=0$$

Add -75 and 91 to get 16.

$$96x^{2}-140x+16=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 96 for a, -140 for b, and 16 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-140\right)±\sqrt{\left(-140\right)^{2}-4\times 96\times 16}}{2\times 96}$$

Square -140.

$$x=\frac{-\left(-140\right)±\sqrt{19600-4\times 96\times 16}}{2\times 96}$$

Multiply -4 times 96.

$$x=\frac{-\left(-140\right)±\sqrt{19600-384\times 16}}{2\times 96}$$

Multiply -384 times 16.

$$x=\frac{-\left(-140\right)±\sqrt{19600-6144}}{2\times 96}$$

Add 19600 to -6144.

$$x=\frac{-\left(-140\right)±\sqrt{13456}}{2\times 96}$$

Take the square root of 13456.

$$x=\frac{-\left(-140\right)±116}{2\times 96}$$

The opposite of -140 is 140.

$$x=\frac{140±116}{2\times 96}$$

Multiply 2 times 96.

$$x=\frac{140±116}{192}$$

Now solve the equation x=\frac{140±116}{192} when ± is plus. Add 140 to 116.

$$x=\frac{256}{192}$$

Reduce the fraction \frac{256}{192} to lowest terms by extracting and canceling out 64.

$$x=\frac{4}{3}$$

Now solve the equation x=\frac{140±116}{192} when ± is minus. Subtract 116 from 140.

$$x=\frac{24}{192}$$

Reduce the fraction \frac{24}{192} to lowest terms by extracting and canceling out 24.

$$x=\frac{1}{8}$$

The equation is now solved.

$$x=\frac{4}{3} x=\frac{1}{8}$$

Steps for Completing the Square

Use the distributive property to multiply 8x-15 by 12x+5 and combine like terms.

$$96x^{2}-140x-75=-91$$

Add 75 to both sides.

$$96x^{2}-140x=-91+75$$

Add -91 and 75 to get -16.

$$96x^{2}-140x=-16$$

Divide both sides by 96.

$$\frac{96x^{2}-140x}{96}=-\frac{16}{96}$$

Dividing by 96 undoes the multiplication by 96.

$$x^{2}+\left(-\frac{140}{96}\right)x=-\frac{16}{96}$$

Reduce the fraction \frac{-140}{96} to lowest terms by extracting and canceling out 4.

$$x^{2}-\frac{35}{24}x=-\frac{16}{96}$$

Reduce the fraction \frac{-16}{96} to lowest terms by extracting and canceling out 16.

$$x^{2}-\frac{35}{24}x=-\frac{1}{6}$$

Divide -\frac{35}{24}, the coefficient of the x term, by 2 to get -\frac{35}{48}. Then add the square of -\frac{35}{48} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{35}{24}x+\left(-\frac{35}{48}\right)^{2}=-\frac{1}{6}+\left(-\frac{35}{48}\right)^{2}$$

Square -\frac{35}{48} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{35}{24}x+\frac{1225}{2304}=-\frac{1}{6}+\frac{1225}{2304}$$

Add -\frac{1}{6} to \frac{1225}{2304} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{35}{24}x+\frac{1225}{2304}=\frac{841}{2304}$$

Factor x^{2}-\frac{35}{24}x+\frac{1225}{2304}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{35}{48}\right)^{2}=\frac{841}{2304}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{35}{48}\right)^{2}}=\sqrt{\frac{841}{2304}}$$

Simplify.

$$x-\frac{35}{48}=\frac{29}{48} x-\frac{35}{48}=-\frac{29}{48}$$

Add \frac{35}{48} to both sides of the equation.

$$x=\frac{4}{3} x=\frac{1}{8}$$