Solve x^2-25x+156

$${ x }^{ 2 } -25x+156$$

Factor

$$\left(x-13\right)\left(x-12\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+156. To find a and b, set up a system to be solved.

$$a+b=-25 ab=1\times 156=156$$

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 156.

$$-1,-156 -2,-78 -3,-52 -4,-39 -6,-26 -12,-13$$

Calculate the sum for each pair.

$$-1-156=-157 -2-78=-80 -3-52=-55 -4-39=-43 -6-26=-32 -12-13=-25$$

The solution is the pair that gives sum -25.

$$a=-13 b=-12$$

Rewrite x^{2}-25x+156 as \left(x^{2}-13x\right)+\left(-12x+156\right).

$$\left(x^{2}-13x\right)+\left(-12x+156\right)$$

Factor out x in the first and -12 in the second group.

$$x\left(x-13\right)-12\left(x-13\right)$$

Factor out common term x-13 by using distributive property.

$$\left(x-13\right)\left(x-12\right)$$

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.

$$x^{2}-25x+156=0$$

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.

$$x=\frac{-\left(-25\right)±\sqrt{\left(-25\right)^{2}-4\times 156}}{2}$$

Square -25.

$$x=\frac{-\left(-25\right)±\sqrt{625-4\times 156}}{2}$$

Multiply -4 times 156.

$$x=\frac{-\left(-25\right)±\sqrt{625-624}}{2}$$

Add 625 to -624.

$$x=\frac{-\left(-25\right)±\sqrt{1}}{2}$$

Take the square root of 1.

$$x=\frac{-\left(-25\right)±1}{2}$$

The opposite of -25 is 25.

$$x=\frac{25±1}{2}$$

Now solve the equation x=\frac{25±1}{2} when ± is plus. Add 25 to 1.

$$x=\frac{26}{2}$$

Divide 26 by 2.

$$x=13$$

Now solve the equation x=\frac{25±1}{2} when ± is minus. Subtract 1 from 25.

$$x=\frac{24}{2}$$

Divide 24 by 2.

$$x=12$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 13 for x_{1} and 12 for x_{2}.

$$x^{2}-25x+156=\left(x-13\right)\left(x-12\right)$$

Evaluate

$$\left(x-13\right)\left(x-12\right)$$