Solve -16+72x-81x^2=

$$- 16 + 72 x - 81 x ^ { 2 } =$$

Factor

$$-\left(9x-4\right)^{2}$$

Solution Steps

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

$$-81x^{2}+72x-16$$

Factor the expression by grouping. First, the expression needs to be rewritten as -81x^{2}+ax+bx-16. To find a and b, set up a system to be solved.

$$a+b=72 ab=-81\left(-16\right)=1296$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 1296.

$$1,1296 2,648 3,432 4,324 6,216 8,162 9,144 12,108 16,81 18,72 24,54 27,48 36,36$$

Calculate the sum for each pair.

$$1+1296=1297 2+648=650 3+432=435 4+324=328 6+216=222 8+162=170 9+144=153 12+108=120 16+81=97 18+72=90 24+54=78 27+48=75 36+36=72$$

The solution is the pair that gives sum 72.

$$a=36 b=36$$

Rewrite -81x^{2}+72x-16 as \left(-81x^{2}+36x\right)+\left(36x-16\right).

$$\left(-81x^{2}+36x\right)+\left(36x-16\right)$$

Factor out -9x in the first and 4 in the second group.

$$-9x\left(9x-4\right)+4\left(9x-4\right)$$

Factor out common term 9x-4 by using distributive property.

$$\left(9x-4\right)\left(-9x+4\right)$$

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.

$$-81x^{2}+72x-16=0$$

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.

$$x=\frac{-72±\sqrt{72^{2}-4\left(-81\right)\left(-16\right)}}{2\left(-81\right)}$$

Square 72.

$$x=\frac{-72±\sqrt{5184-4\left(-81\right)\left(-16\right)}}{2\left(-81\right)}$$

Multiply -4 times -81.

$$x=\frac{-72±\sqrt{5184+324\left(-16\right)}}{2\left(-81\right)}$$

Multiply 324 times -16.

$$x=\frac{-72±\sqrt{5184-5184}}{2\left(-81\right)}$$

Add 5184 to -5184.

$$x=\frac{-72±\sqrt{0}}{2\left(-81\right)}$$

Take the square root of 0.

$$x=\frac{-72±0}{2\left(-81\right)}$$

Multiply 2 times -81.

$$x=\frac{-72±0}{-162}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{4}{9} for x_{1} and \frac{4}{9} for x_{2}.

$$-81x^{2}+72x-16=-81\left(x-\frac{4}{9}\right)\left(x-\frac{4}{9}\right)$$

Subtract \frac{4}{9} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$-81x^{2}+72x-16=-81\times \frac{-9x+4}{-9}\left(x-\frac{4}{9}\right)$$

Subtract \frac{4}{9} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$-81x^{2}+72x-16=-81\times \frac{-9x+4}{-9}\times \frac{-9x+4}{-9}$$

Multiply \frac{-9x+4}{-9} times \frac{-9x+4}{-9} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$-81x^{2}+72x-16=-81\times \frac{\left(-9x+4\right)\left(-9x+4\right)}{-9\left(-9\right)}$$

Multiply -9 times -9.

$$-81x^{2}+72x-16=-81\times \frac{\left(-9x+4\right)\left(-9x+4\right)}{81}$$

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

$$-81x^{2}+72x-16=-\left(-9x+4\right)\left(-9x+4\right)$$

Evaluate

$$-\left(9x-4\right)^{2}$$