Solve 11x^2-60x+64

$$11 { x }^{ 2 } -60x+64$$

Factor

$$\left(x-4\right)\left(11x-16\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 11x^{2}+ax+bx+64. To find a and b, set up a system to be solved.

$$a+b=-60 ab=11\times 64=704$$

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 704.

$$-1,-704 -2,-352 -4,-176 -8,-88 -11,-64 -16,-44 -22,-32$$

Calculate the sum for each pair.

$$-1-704=-705 -2-352=-354 -4-176=-180 -8-88=-96 -11-64=-75 -16-44=-60 -22-32=-54$$

The solution is the pair that gives sum -60.

$$a=-44 b=-16$$

Rewrite 11x^{2}-60x+64 as \left(11x^{2}-44x\right)+\left(-16x+64\right).

$$\left(11x^{2}-44x\right)+\left(-16x+64\right)$$

Factor out 11x in the first and -16 in the second group.

$$11x\left(x-4\right)-16\left(x-4\right)$$

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

$$\left(x-4\right)\left(11x-16\right)$$

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.

$$11x^{2}-60x+64=0$$

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.

$$x=\frac{-\left(-60\right)±\sqrt{\left(-60\right)^{2}-4\times 11\times 64}}{2\times 11}$$

Square -60.

$$x=\frac{-\left(-60\right)±\sqrt{3600-4\times 11\times 64}}{2\times 11}$$

Multiply -4 times 11.

$$x=\frac{-\left(-60\right)±\sqrt{3600-44\times 64}}{2\times 11}$$

Multiply -44 times 64.

$$x=\frac{-\left(-60\right)±\sqrt{3600-2816}}{2\times 11}$$

Add 3600 to -2816.

$$x=\frac{-\left(-60\right)±\sqrt{784}}{2\times 11}$$

Take the square root of 784.

$$x=\frac{-\left(-60\right)±28}{2\times 11}$$

The opposite of -60 is 60.

$$x=\frac{60±28}{2\times 11}$$

Multiply 2 times 11.

$$x=\frac{60±28}{22}$$

Now solve the equation x=\frac{60±28}{22} when ± is plus. Add 60 to 28.

$$x=\frac{88}{22}$$

Divide 88 by 22.

$$x=4$$

Now solve the equation x=\frac{60±28}{22} when ± is minus. Subtract 28 from 60.

$$x=\frac{32}{22}$$

Reduce the fraction \frac{32}{22} to lowest terms by extracting and canceling out 2.

$$x=\frac{16}{11}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 4 for x_{1} and \frac{16}{11} for x_{2}.

$$11x^{2}-60x+64=11\left(x-4\right)\left(x-\frac{16}{11}\right)$$

Subtract \frac{16}{11} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$11x^{2}-60x+64=11\left(x-4\right)\times \frac{11x-16}{11}$$

Cancel out 11, the greatest common factor in 11 and 11.

$$11x^{2}-60x+64=\left(x-4\right)\left(11x-16\right)$$

Evaluate

$$\left(x-4\right)\left(11x-16\right)$$