Solve 11x^2-369x+2158=

$$11 { x }^{ 2 } -369x+2158=$$

Factor

$$\left(x-26\right)\left(11x-83\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 11x^{2}+ax+bx+2158. To find a and b, set up a system to be solved.

$$a+b=-369 ab=11\times 2158=23738$$

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 23738.

$$-1,-23738 -2,-11869 -11,-2158 -13,-1826 -22,-1079 -26,-913 -83,-286 -143,-166$$

Calculate the sum for each pair.

$$-1-23738=-23739 -2-11869=-11871 -11-2158=-2169 -13-1826=-1839 -22-1079=-1101 -26-913=-939 -83-286=-369 -143-166=-309$$

The solution is the pair that gives sum -369.

$$a=-286 b=-83$$

Rewrite 11x^{2}-369x+2158 as \left(11x^{2}-286x\right)+\left(-83x+2158\right).

$$\left(11x^{2}-286x\right)+\left(-83x+2158\right)$$

Factor out 11x in the first and -83 in the second group.

$$11x\left(x-26\right)-83\left(x-26\right)$$

Factor out common term x-26 by using distributive property.

$$\left(x-26\right)\left(11x-83\right)$$

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.

$$11x^{2}-369x+2158=0$$

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.

$$x=\frac{-\left(-369\right)±\sqrt{\left(-369\right)^{2}-4\times 11\times 2158}}{2\times 11}$$

Square -369.

$$x=\frac{-\left(-369\right)±\sqrt{136161-4\times 11\times 2158}}{2\times 11}$$

Multiply -4 times 11.

$$x=\frac{-\left(-369\right)±\sqrt{136161-44\times 2158}}{2\times 11}$$

Multiply -44 times 2158.

$$x=\frac{-\left(-369\right)±\sqrt{136161-94952}}{2\times 11}$$

Add 136161 to -94952.

$$x=\frac{-\left(-369\right)±\sqrt{41209}}{2\times 11}$$

Take the square root of 41209.

$$x=\frac{-\left(-369\right)±203}{2\times 11}$$

The opposite of -369 is 369.

$$x=\frac{369±203}{2\times 11}$$

Multiply 2 times 11.

$$x=\frac{369±203}{22}$$

Now solve the equation x=\frac{369±203}{22} when ± is plus. Add 369 to 203.

$$x=\frac{572}{22}$$

Divide 572 by 22.

$$x=26$$

Now solve the equation x=\frac{369±203}{22} when ± is minus. Subtract 203 from 369.

$$x=\frac{166}{22}$$

Reduce the fraction \frac{166}{22} to lowest terms by extracting and canceling out 2.

$$x=\frac{83}{11}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 26 for x_{1} and \frac{83}{11} for x_{2}.

$$11x^{2}-369x+2158=11\left(x-26\right)\left(x-\frac{83}{11}\right)$$

Subtract \frac{83}{11} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$11x^{2}-369x+2158=11\left(x-26\right)\times \frac{11x-83}{11}$$

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

$$11x^{2}-369x+2158=\left(x-26\right)\left(11x-83\right)$$

Evaluate

$$\left(x-26\right)\left(11x-83\right)$$