Solve -72x^2-71x-7

$$-72 { x }^{ 2 } -71x-7$$

Factor

$$-\left(8x+7\right)\left(9x+1\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as -72x^{2}+ax+bx-7. To find a and b, set up a system to be solved.

$$a+b=-71 ab=-72\left(-7\right)=504$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 504.

$$-1,-504 -2,-252 -3,-168 -4,-126 -6,-84 -7,-72 -8,-63 -9,-56 -12,-42 -14,-36 -18,-28 -21,-24$$

Calculate the sum for each pair.

$$-1-504=-505 -2-252=-254 -3-168=-171 -4-126=-130 -6-84=-90 -7-72=-79 -8-63=-71 -9-56=-65 -12-42=-54 -14-36=-50 -18-28=-46 -21-24=-45$$

The solution is the pair that gives sum -71.

$$a=-8 b=-63$$

Rewrite -72x^{2}-71x-7 as \left(-72x^{2}-8x\right)+\left(-63x-7\right).

$$\left(-72x^{2}-8x\right)+\left(-63x-7\right)$$

Factor out 8x in the first and 7 in the second group.

$$8x\left(-9x-1\right)+7\left(-9x-1\right)$$

Factor out common term -9x-1 by using distributive property.

$$\left(-9x-1\right)\left(8x+7\right)$$

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.

$$-72x^{2}-71x-7=0$$

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.

$$x=\frac{-\left(-71\right)±\sqrt{\left(-71\right)^{2}-4\left(-72\right)\left(-7\right)}}{2\left(-72\right)}$$

Square -71.

$$x=\frac{-\left(-71\right)±\sqrt{5041-4\left(-72\right)\left(-7\right)}}{2\left(-72\right)}$$

Multiply -4 times -72.

$$x=\frac{-\left(-71\right)±\sqrt{5041+288\left(-7\right)}}{2\left(-72\right)}$$

Multiply 288 times -7.

$$x=\frac{-\left(-71\right)±\sqrt{5041-2016}}{2\left(-72\right)}$$

Add 5041 to -2016.

$$x=\frac{-\left(-71\right)±\sqrt{3025}}{2\left(-72\right)}$$

Take the square root of 3025.

$$x=\frac{-\left(-71\right)±55}{2\left(-72\right)}$$

The opposite of -71 is 71.

$$x=\frac{71±55}{2\left(-72\right)}$$

Multiply 2 times -72.

$$x=\frac{71±55}{-144}$$

Now solve the equation x=\frac{71±55}{-144} when ± is plus. Add 71 to 55.

$$x=\frac{126}{-144}$$

Reduce the fraction \frac{126}{-144} to lowest terms by extracting and canceling out 18.

$$x=-\frac{7}{8}$$

Now solve the equation x=\frac{71±55}{-144} when ± is minus. Subtract 55 from 71.

$$x=\frac{16}{-144}$$

Reduce the fraction \frac{16}{-144} to lowest terms by extracting and canceling out 16.

$$x=-\frac{1}{9}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{7}{8} for x_{1} and -\frac{1}{9} for x_{2}.

$$-72x^{2}-71x-7=-72\left(x-\left(-\frac{7}{8}\right)\right)\left(x-\left(-\frac{1}{9}\right)\right)$$

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

$$-72x^{2}-71x-7=-72\left(x+\frac{7}{8}\right)\left(x+\frac{1}{9}\right)$$

Add \frac{7}{8} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$-72x^{2}-71x-7=-72\times \frac{-8x-7}{-8}\left(x+\frac{1}{9}\right)$$

Add \frac{1}{9} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$-72x^{2}-71x-7=-72\times \frac{-8x-7}{-8}\times \frac{-9x-1}{-9}$$

Multiply \frac{-8x-7}{-8} times \frac{-9x-1}{-9} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$-72x^{2}-71x-7=-72\times \frac{\left(-8x-7\right)\left(-9x-1\right)}{-8\left(-9\right)}$$

Multiply -8 times -9.

$$-72x^{2}-71x-7=-72\times \frac{\left(-8x-7\right)\left(-9x-1\right)}{72}$$

Cancel out 72, the greatest common factor in -72 and 72.

$$-72x^{2}-71x-7=-\left(-8x-7\right)\left(-9x-1\right)$$

Evaluate

$$-72x^{2}-71x-7$$