Solve x^2-52x+100

$${ x }^{ 2 } -52x+100$$

Factor

$$\left(x-50\right)\left(x-2\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+100. To find a and b, set up a system to be solved.

$$a+b=-52 ab=1\times 100=100$$

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 100.

$$-1,-100 -2,-50 -4,-25 -5,-20 -10,-10$$

Calculate the sum for each pair.

$$-1-100=-101 -2-50=-52 -4-25=-29 -5-20=-25 -10-10=-20$$

The solution is the pair that gives sum -52.

$$a=-50 b=-2$$

Rewrite x^{2}-52x+100 as \left(x^{2}-50x\right)+\left(-2x+100\right).

$$\left(x^{2}-50x\right)+\left(-2x+100\right)$$

Factor out x in the first and -2 in the second group.

$$x\left(x-50\right)-2\left(x-50\right)$$

Factor out common term x-50 by using distributive property.

$$\left(x-50\right)\left(x-2\right)$$

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.

$$x^{2}-52x+100=0$$

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.

$$x=\frac{-\left(-52\right)±\sqrt{\left(-52\right)^{2}-4\times 100}}{2}$$

Square -52.

$$x=\frac{-\left(-52\right)±\sqrt{2704-4\times 100}}{2}$$

Multiply -4 times 100.

$$x=\frac{-\left(-52\right)±\sqrt{2704-400}}{2}$$

Add 2704 to -400.

$$x=\frac{-\left(-52\right)±\sqrt{2304}}{2}$$

Take the square root of 2304.

$$x=\frac{-\left(-52\right)±48}{2}$$

The opposite of -52 is 52.

$$x=\frac{52±48}{2}$$

Now solve the equation x=\frac{52±48}{2} when ± is plus. Add 52 to 48.

$$x=\frac{100}{2}$$

Divide 100 by 2.

$$x=50$$

Now solve the equation x=\frac{52±48}{2} when ± is minus. Subtract 48 from 52.

$$x=\frac{4}{2}$$

Divide 4 by 2.

$$x=2$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 50 for x_{1} and 2 for x_{2}.

$$x^{2}-52x+100=\left(x-50\right)\left(x-2\right)$$

Evaluate

$$\left(x-50\right)\left(x-2\right)$$