Solve x^2+11x+10

$${ x }^{ 2 } +11x+10$$

Factor

$$\left(x+1\right)\left(x+10\right)$$

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.

$$a+b=11 ab=1\times 10=10$$

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 11.

$$a=1 b=10$$

Rewrite x^{2}+11x+10 as \left(x^{2}+x\right)+\left(10x+10\right).

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

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

$$x\left(x+1\right)+10\left(x+1\right)$$

Factor out common term x+1 by using distributive property.

$$\left(x+1\right)\left(x+10\right)$$

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.

$$x^{2}+11x+10=0$$

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.

$$x=\frac{-11±\sqrt{11^{2}-4\times 10}}{2}$$

Square 11.

$$x=\frac{-11±\sqrt{121-4\times 10}}{2}$$

Multiply -4 times 10.

$$x=\frac{-11±\sqrt{121-40}}{2}$$

Add 121 to -40.

$$x=\frac{-11±\sqrt{81}}{2}$$

Take the square root of 81.

$$x=\frac{-11±9}{2}$$

Now solve the equation x=\frac{-11±9}{2} when ± is plus. Add -11 to 9.

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

Divide -2 by 2.

$$x=-1$$

Now solve the equation x=\frac{-11±9}{2} when ± is minus. Subtract 9 from -11.

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

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 -1 for x_{1} and -10 for x_{2}.

$$x^{2}+11x+10=\left(x-\left(-1\right)\right)\left(x-\left(-10\right)\right)$$

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

$$x^{2}+11x+10=\left(x+1\right)\left(x+10\right)$$

Evaluate

$$\left(x+1\right)\left(x+10\right)$$