Solve x^2+15x+50

$${ x }^{ 2 } +15x+50$$

Factor

$$\left(x+5\right)\left(x+10\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+50. To find a and b, set up a system to be solved.

$$a+b=15 ab=1\times 50=50$$

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 50.

$$1,50 2,25 5,10$$

Calculate the sum for each pair.

$$1+50=51 2+25=27 5+10=15$$

The solution is the pair that gives sum 15.

$$a=5 b=10$$

Rewrite x^{2}+15x+50 as \left(x^{2}+5x\right)+\left(10x+50\right).

$$\left(x^{2}+5x\right)+\left(10x+50\right)$$

Factor out x in the first and 10 in the second group.

$$x\left(x+5\right)+10\left(x+5\right)$$

Factor out common term x+5 by using distributive property.

$$\left(x+5\right)\left(x+10\right)$$

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.

$$x^{2}+15x+50=0$$

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.

$$x=\frac{-15±\sqrt{15^{2}-4\times 50}}{2}$$

Square 15.

$$x=\frac{-15±\sqrt{225-4\times 50}}{2}$$

Multiply -4 times 50.

$$x=\frac{-15±\sqrt{225-200}}{2}$$

Add 225 to -200.

$$x=\frac{-15±\sqrt{25}}{2}$$

Take the square root of 25.

$$x=\frac{-15±5}{2}$$

Now solve the equation x=\frac{-15±5}{2} when ± is plus. Add -15 to 5.

$$x=-\frac{10}{2}$$

Divide -10 by 2.

$$x=-5$$

Now solve the equation x=\frac{-15±5}{2} when ± is minus. Subtract 5 from -15.

$$x=-\frac{20}{2}$$

Divide -20 by 2.

$$x=-10$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -5 for x_{1} and -10 for x_{2}.

$$x^{2}+15x+50=\left(x-\left(-5\right)\right)\left(x-\left(-10\right)\right)$$

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

$$x^{2}+15x+50=\left(x+5\right)\left(x+10\right)$$

Evaluate

$$\left(x+5\right)\left(x+10\right)$$