Solve x^2+52x+100

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

Factor

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

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 positive, a and b are both positive. 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.

$$a=2 b=50$$

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

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

Factor out x in the first and 50 in the second group.

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

Factor out common term x+2 by using distributive property.

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

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.

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

Square 52.

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

Multiply -4 times 100.

$$x=\frac{-52±\sqrt{2704-400}}{2}$$

Add 2704 to -400.

$$x=\frac{-52±\sqrt{2304}}{2}$$

Take the square root of 2304.

$$x=\frac{-52±48}{2}$$

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

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

Divide -4 by 2.

$$x=-2$$

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

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

Divide -100 by 2.

$$x=-50$$

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

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

Simplify all the expressions of the form p-\left(-q\right) to p+q.

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

Evaluate

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