Solve -117x^2+477x-36

$$-117 { x }^{ 2 } +477x-36$$

Factor

$$9\left(4-x\right)\left(13x-1\right)$$

Solution Steps

Factor out 9.

$$9\left(-13x^{2}+53x-4\right)$$

Consider -13x^{2}+53x-4. Factor the expression by grouping. First, the expression needs to be rewritten as -13x^{2}+ax+bx-4. To find a and b, set up a system to be solved.

$$a+b=53 ab=-13\left(-4\right)=52$$

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 52.

$$1,52 2,26 4,13$$

Calculate the sum for each pair.

$$1+52=53 2+26=28 4+13=17$$

The solution is the pair that gives sum 53.

$$a=52 b=1$$

Rewrite -13x^{2}+53x-4 as \left(-13x^{2}+52x\right)+\left(x-4\right).

$$\left(-13x^{2}+52x\right)+\left(x-4\right)$$

Factor out 13x in the first and -1 in the second group.

$$13x\left(-x+4\right)-\left(-x+4\right)$$

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

$$\left(-x+4\right)\left(13x-1\right)$$

Rewrite the complete factored expression.

$$9\left(-x+4\right)\left(13x-1\right)$$

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.

$$-117x^{2}+477x-36=0$$

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.

$$x=\frac{-477±\sqrt{477^{2}-4\left(-117\right)\left(-36\right)}}{2\left(-117\right)}$$

Square 477.

$$x=\frac{-477±\sqrt{227529-4\left(-117\right)\left(-36\right)}}{2\left(-117\right)}$$

Multiply -4 times -117.

$$x=\frac{-477±\sqrt{227529+468\left(-36\right)}}{2\left(-117\right)}$$

Multiply 468 times -36.

$$x=\frac{-477±\sqrt{227529-16848}}{2\left(-117\right)}$$

Add 227529 to -16848.

$$x=\frac{-477±\sqrt{210681}}{2\left(-117\right)}$$

Take the square root of 210681.

$$x=\frac{-477±459}{2\left(-117\right)}$$

Multiply 2 times -117.

$$x=\frac{-477±459}{-234}$$

Now solve the equation x=\frac{-477±459}{-234} when ± is plus. Add -477 to 459.

$$x=-\frac{18}{-234}$$

Reduce the fraction \frac{-18}{-234} to lowest terms by extracting and canceling out 18.

$$x=\frac{1}{13}$$

Now solve the equation x=\frac{-477±459}{-234} when ± is minus. Subtract 459 from -477.

$$x=-\frac{936}{-234}$$

Divide -936 by -234.

$$x=4$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{1}{13} for x_{1} and 4 for x_{2}.

$$-117x^{2}+477x-36=-117\left(x-\frac{1}{13}\right)\left(x-4\right)$$

Subtract \frac{1}{13} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$-117x^{2}+477x-36=-117\times \frac{-13x+1}{-13}\left(x-4\right)$$

Cancel out 13, the greatest common factor in -117 and 13.

$$-117x^{2}+477x-36=9\left(-13x+1\right)\left(x-4\right)$$

Evaluate

$$-117x^{2}+477x-36$$