Solve 15x^2-8x-16

$$15 { x }^{ 2 } -8x-16$$

Factor

$$\left(3x-4\right)\left(5x+4\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 15x^{2}+ax+bx-16. To find a and b, set up a system to be solved.

$$a+b=-8 ab=15\left(-16\right)=-240$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -240.

$$1,-240 2,-120 3,-80 4,-60 5,-48 6,-40 8,-30 10,-24 12,-20 15,-16$$

Calculate the sum for each pair.

$$1-240=-239 2-120=-118 3-80=-77 4-60=-56 5-48=-43 6-40=-34 8-30=-22 10-24=-14 12-20=-8 15-16=-1$$

The solution is the pair that gives sum -8.

$$a=-20 b=12$$

Rewrite 15x^{2}-8x-16 as \left(15x^{2}-20x\right)+\left(12x-16\right).

$$\left(15x^{2}-20x\right)+\left(12x-16\right)$$

Factor out 5x in the first and 4 in the second group.

$$5x\left(3x-4\right)+4\left(3x-4\right)$$

Factor out common term 3x-4 by using distributive property.

$$\left(3x-4\right)\left(5x+4\right)$$

Steps Using the Quadratic Formula

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$15x^{2}-8x-16=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 15\left(-16\right)}}{2\times 15}$$

Square -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\times 15\left(-16\right)}}{2\times 15}$$

Multiply -4 times 15.

$$x=\frac{-\left(-8\right)±\sqrt{64-60\left(-16\right)}}{2\times 15}$$

Multiply -60 times -16.

$$x=\frac{-\left(-8\right)±\sqrt{64+960}}{2\times 15}$$

Add 64 to 960.

$$x=\frac{-\left(-8\right)±\sqrt{1024}}{2\times 15}$$

Take the square root of 1024.

$$x=\frac{-\left(-8\right)±32}{2\times 15}$$

The opposite of -8 is 8.

$$x=\frac{8±32}{2\times 15}$$

Multiply 2 times 15.

$$x=\frac{8±32}{30}$$

Now solve the equation x=\frac{8±32}{30} when ± is plus. Add 8 to 32.

$$x=\frac{40}{30}$$

Reduce the fraction \frac{40}{30} to lowest terms by extracting and canceling out 10.

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

Now solve the equation x=\frac{8±32}{30} when ± is minus. Subtract 32 from 8.

$$x=-\frac{24}{30}$$

Reduce the fraction \frac{-24}{30} to lowest terms by extracting and canceling out 6.

$$x=-\frac{4}{5}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{4}{3} for x_{1} and -\frac{4}{5} for x_{2}.

$$15x^{2}-8x-16=15\left(x-\frac{4}{3}\right)\left(x-\left(-\frac{4}{5}\right)\right)$$

Simplify all the expressions of the form p-\left(-q\right) to p+q.

$$15x^{2}-8x-16=15\left(x-\frac{4}{3}\right)\left(x+\frac{4}{5}\right)$$

Subtract \frac{4}{3} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$15x^{2}-8x-16=15\times \frac{3x-4}{3}\left(x+\frac{4}{5}\right)$$

Add \frac{4}{5} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$15x^{2}-8x-16=15\times \frac{3x-4}{3}\times \frac{5x+4}{5}$$

Multiply \frac{3x-4}{3} times \frac{5x+4}{5} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$15x^{2}-8x-16=15\times \frac{\left(3x-4\right)\left(5x+4\right)}{3\times 5}$$

Multiply 3 times 5.

$$15x^{2}-8x-16=15\times \frac{\left(3x-4\right)\left(5x+4\right)}{15}$$

Cancel out 15, the greatest common factor in 15 and 15.

$$15x^{2}-8x-16=\left(3x-4\right)\left(5x+4\right)$$

Evaluate

$$\left(3x-4\right)\left(5x+4\right)$$