Solve 10x^2-107x+187

$$10 { x }^{ 2 } -107x+187$$

Factor

$$\left(2x-17\right)\left(5x-11\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 10x^{2}+ax+bx+187. To find a and b, set up a system to be solved.

$$a+b=-107 ab=10\times 187=1870$$

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 1870.

$$-1,-1870 -2,-935 -5,-374 -10,-187 -11,-170 -17,-110 -22,-85 -34,-55$$

Calculate the sum for each pair.

$$-1-1870=-1871 -2-935=-937 -5-374=-379 -10-187=-197 -11-170=-181 -17-110=-127 -22-85=-107 -34-55=-89$$

The solution is the pair that gives sum -107.

$$a=-85 b=-22$$

Rewrite 10x^{2}-107x+187 as \left(10x^{2}-85x\right)+\left(-22x+187\right).

$$\left(10x^{2}-85x\right)+\left(-22x+187\right)$$

Factor out 5x in the first and -11 in the second group.

$$5x\left(2x-17\right)-11\left(2x-17\right)$$

Factor out common term 2x-17 by using distributive property.

$$\left(2x-17\right)\left(5x-11\right)$$

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.

$$10x^{2}-107x+187=0$$

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.

$$x=\frac{-\left(-107\right)±\sqrt{\left(-107\right)^{2}-4\times 10\times 187}}{2\times 10}$$

Square -107.

$$x=\frac{-\left(-107\right)±\sqrt{11449-4\times 10\times 187}}{2\times 10}$$

Multiply -4 times 10.

$$x=\frac{-\left(-107\right)±\sqrt{11449-40\times 187}}{2\times 10}$$

Multiply -40 times 187.

$$x=\frac{-\left(-107\right)±\sqrt{11449-7480}}{2\times 10}$$

Add 11449 to -7480.

$$x=\frac{-\left(-107\right)±\sqrt{3969}}{2\times 10}$$

Take the square root of 3969.

$$x=\frac{-\left(-107\right)±63}{2\times 10}$$

The opposite of -107 is 107.

$$x=\frac{107±63}{2\times 10}$$

Multiply 2 times 10.

$$x=\frac{107±63}{20}$$

Now solve the equation x=\frac{107±63}{20} when ± is plus. Add 107 to 63.

$$x=\frac{170}{20}$$

Reduce the fraction \frac{170}{20} to lowest terms by extracting and canceling out 10.

$$x=\frac{17}{2}$$

Now solve the equation x=\frac{107±63}{20} when ± is minus. Subtract 63 from 107.

$$x=\frac{44}{20}$$

Reduce the fraction \frac{44}{20} to lowest terms by extracting and canceling out 4.

$$x=\frac{11}{5}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{17}{2} for x_{1} and \frac{11}{5} for x_{2}.

$$10x^{2}-107x+187=10\left(x-\frac{17}{2}\right)\left(x-\frac{11}{5}\right)$$

Subtract \frac{17}{2} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$10x^{2}-107x+187=10\times \frac{2x-17}{2}\left(x-\frac{11}{5}\right)$$

Subtract \frac{11}{5} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$10x^{2}-107x+187=10\times \frac{2x-17}{2}\times \frac{5x-11}{5}$$

Multiply \frac{2x-17}{2} times \frac{5x-11}{5} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$10x^{2}-107x+187=10\times \frac{\left(2x-17\right)\left(5x-11\right)}{2\times 5}$$

Multiply 2 times 5.

$$10x^{2}-107x+187=10\times \frac{\left(2x-17\right)\left(5x-11\right)}{10}$$

Cancel out 10, the greatest common factor in 10 and 10.

$$10x^{2}-107x+187=\left(2x-17\right)\left(5x-11\right)$$

Evaluate

$$\left(2x-17\right)\left(5x-11\right)$$