Solve x^2+7x+10=

$${ x }^{ 2 } +7x+10=$$

Factor

$$\left(x+2\right)\left(x+5\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+10. To find a and b, set up a system to be solved.

$$a+b=7 ab=1\times 10=10$$

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 10.

$$1,10 2,5$$

Calculate the sum for each pair.

$$1+10=11 2+5=7$$

The solution is the pair that gives sum 7.

$$a=2 b=5$$

Rewrite x^{2}+7x+10 as \left(x^{2}+2x\right)+\left(5x+10\right).

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

Factor out x in the first and 5 in the second group.

$$x\left(x+2\right)+5\left(x+2\right)$$

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

$$\left(x+2\right)\left(x+5\right)$$

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.

$$x^{2}+7x+10=0$$

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.

$$x=\frac{-7±\sqrt{7^{2}-4\times 10}}{2}$$

Square 7.

$$x=\frac{-7±\sqrt{49-4\times 10}}{2}$$

Multiply -4 times 10.

$$x=\frac{-7±\sqrt{49-40}}{2}$$

Add 49 to -40.

$$x=\frac{-7±\sqrt{9}}{2}$$

Take the square root of 9.

$$x=\frac{-7±3}{2}$$

Now solve the equation x=\frac{-7±3}{2} when ± is plus. Add -7 to 3.

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

Divide -4 by 2.

$$x=-2$$

Now solve the equation x=\frac{-7±3}{2} when ± is minus. Subtract 3 from -7.

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

Divide -10 by 2.

$$x=-5$$

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 -5 for x_{2}.

$$x^{2}+7x+10=\left(x-\left(-2\right)\right)\left(x-\left(-5\right)\right)$$

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

$$x^{2}+7x+10=\left(x+2\right)\left(x+5\right)$$

Evaluate

$$\left(x+2\right)\left(x+5\right)$$