Solve -64-64x+9x^2

$$- 64 - 64 x + 9 x ^ { 2 }$$

Factor

$$\left(x-8\right)\left(9x+8\right)$$

Solution Steps

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$9x^{2}-64x-64$$

Factor the expression by grouping. First, the expression needs to be rewritten as 9x^{2}+ax+bx-64. To find a and b, set up a system to be solved.

$$a+b=-64 ab=9\left(-64\right)=-576$$

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 -576.

$$1,-576 2,-288 3,-192 4,-144 6,-96 8,-72 9,-64 12,-48 16,-36 18,-32 24,-24$$

Calculate the sum for each pair.

$$1-576=-575 2-288=-286 3-192=-189 4-144=-140 6-96=-90 8-72=-64 9-64=-55 12-48=-36 16-36=-20 18-32=-14 24-24=0$$

The solution is the pair that gives sum -64.

$$a=-72 b=8$$

Rewrite 9x^{2}-64x-64 as \left(9x^{2}-72x\right)+\left(8x-64\right).

$$\left(9x^{2}-72x\right)+\left(8x-64\right)$$

Factor out 9x in the first and 8 in the second group.

$$9x\left(x-8\right)+8\left(x-8\right)$$

Factor out common term x-8 by using distributive property.

$$\left(x-8\right)\left(9x+8\right)$$

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.

$$9x^{2}-64x-64=0$$

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.

$$x=\frac{-\left(-64\right)±\sqrt{\left(-64\right)^{2}-4\times 9\left(-64\right)}}{2\times 9}$$

Square -64.

$$x=\frac{-\left(-64\right)±\sqrt{4096-4\times 9\left(-64\right)}}{2\times 9}$$

Multiply -4 times 9.

$$x=\frac{-\left(-64\right)±\sqrt{4096-36\left(-64\right)}}{2\times 9}$$

Multiply -36 times -64.

$$x=\frac{-\left(-64\right)±\sqrt{4096+2304}}{2\times 9}$$

Add 4096 to 2304.

$$x=\frac{-\left(-64\right)±\sqrt{6400}}{2\times 9}$$

Take the square root of 6400.

$$x=\frac{-\left(-64\right)±80}{2\times 9}$$

The opposite of -64 is 64.

$$x=\frac{64±80}{2\times 9}$$

Multiply 2 times 9.

$$x=\frac{64±80}{18}$$

Now solve the equation x=\frac{64±80}{18} when ± is plus. Add 64 to 80.

$$x=\frac{144}{18}$$

Divide 144 by 18.

$$x=8$$

Now solve the equation x=\frac{64±80}{18} when ± is minus. Subtract 80 from 64.

$$x=-\frac{16}{18}$$

Reduce the fraction \frac{-16}{18} to lowest terms by extracting and canceling out 2.

$$x=-\frac{8}{9}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 8 for x_{1} and -\frac{8}{9} for x_{2}.

$$9x^{2}-64x-64=9\left(x-8\right)\left(x-\left(-\frac{8}{9}\right)\right)$$

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

$$9x^{2}-64x-64=9\left(x-8\right)\left(x+\frac{8}{9}\right)$$

Add \frac{8}{9} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$9x^{2}-64x-64=9\left(x-8\right)\times \frac{9x+8}{9}$$

Cancel out 9, the greatest common factor in 9 and 9.

$$9x^{2}-64x-64=\left(x-8\right)\left(9x+8\right)$$

Evaluate

$$\left(x-8\right)\left(9x+8\right)$$