Solve -81x^2-54x-5

$$-81 { x }^{ 2 } -54x-5$$

Factor

$$-\left(9x+1\right)\left(9x+5\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as -81x^{2}+ax+bx-5. To find a and b, set up a system to be solved.

$$a+b=-54 ab=-81\left(-5\right)=405$$

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 405.

$$-1,-405 -3,-135 -5,-81 -9,-45 -15,-27$$

Calculate the sum for each pair.

$$-1-405=-406 -3-135=-138 -5-81=-86 -9-45=-54 -15-27=-42$$

The solution is the pair that gives sum -54.

$$a=-9 b=-45$$

Rewrite -81x^{2}-54x-5 as \left(-81x^{2}-9x\right)+\left(-45x-5\right).

$$\left(-81x^{2}-9x\right)+\left(-45x-5\right)$$

Factor out 9x in the first and 5 in the second group.

$$9x\left(-9x-1\right)+5\left(-9x-1\right)$$

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

$$\left(-9x-1\right)\left(9x+5\right)$$

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.

$$-81x^{2}-54x-5=0$$

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.

$$x=\frac{-\left(-54\right)±\sqrt{\left(-54\right)^{2}-4\left(-81\right)\left(-5\right)}}{2\left(-81\right)}$$

Square -54.

$$x=\frac{-\left(-54\right)±\sqrt{2916-4\left(-81\right)\left(-5\right)}}{2\left(-81\right)}$$

Multiply -4 times -81.

$$x=\frac{-\left(-54\right)±\sqrt{2916+324\left(-5\right)}}{2\left(-81\right)}$$

Multiply 324 times -5.

$$x=\frac{-\left(-54\right)±\sqrt{2916-1620}}{2\left(-81\right)}$$

Add 2916 to -1620.

$$x=\frac{-\left(-54\right)±\sqrt{1296}}{2\left(-81\right)}$$

Take the square root of 1296.

$$x=\frac{-\left(-54\right)±36}{2\left(-81\right)}$$

The opposite of -54 is 54.

$$x=\frac{54±36}{2\left(-81\right)}$$

Multiply 2 times -81.

$$x=\frac{54±36}{-162}$$

Now solve the equation x=\frac{54±36}{-162} when ± is plus. Add 54 to 36.

$$x=\frac{90}{-162}$$

Reduce the fraction \frac{90}{-162} to lowest terms by extracting and canceling out 18.

$$x=-\frac{5}{9}$$

Now solve the equation x=\frac{54±36}{-162} when ± is minus. Subtract 36 from 54.

$$x=\frac{18}{-162}$$

Reduce the fraction \frac{18}{-162} to lowest terms by extracting and canceling out 18.

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

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{5}{9} for x_{1} and -\frac{1}{9} for x_{2}.

$$-81x^{2}-54x-5=-81\left(x-\left(-\frac{5}{9}\right)\right)\left(x-\left(-\frac{1}{9}\right)\right)$$

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

$$-81x^{2}-54x-5=-81\left(x+\frac{5}{9}\right)\left(x+\frac{1}{9}\right)$$

Add \frac{5}{9} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$-81x^{2}-54x-5=-81\times \frac{-9x-5}{-9}\left(x+\frac{1}{9}\right)$$

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

$$-81x^{2}-54x-5=-81\times \frac{-9x-5}{-9}\times \frac{-9x-1}{-9}$$

Multiply \frac{-9x-5}{-9} times \frac{-9x-1}{-9} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$-81x^{2}-54x-5=-81\times \frac{\left(-9x-5\right)\left(-9x-1\right)}{-9\left(-9\right)}$$

Multiply -9 times -9.

$$-81x^{2}-54x-5=-81\times \frac{\left(-9x-5\right)\left(-9x-1\right)}{81}$$

Cancel out 81, the greatest common factor in -81 and 81.

$$-81x^{2}-54x-5=-\left(-9x-5\right)\left(-9x-1\right)$$

Evaluate

$$-81x^{2}-54x-5$$