Solve 9x^2+11x+2

$$9 { x }^{ 2 } +11x+2$$

Factor

$$\left(x+1\right)\left(9x+2\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 9x^{2}+ax+bx+2. To find a and b, set up a system to be solved.

$$a+b=11 ab=9\times 2=18$$

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 18.

$$1,18 2,9 3,6$$

Calculate the sum for each pair.

$$1+18=19 2+9=11 3+6=9$$

The solution is the pair that gives sum 11.

$$a=2 b=9$$

Rewrite 9x^{2}+11x+2 as \left(9x^{2}+2x\right)+\left(9x+2\right).

$$\left(9x^{2}+2x\right)+\left(9x+2\right)$$

Factor out x in 9x^{2}+2x.

$$x\left(9x+2\right)+9x+2$$

Factor out common term 9x+2 by using distributive property.

$$\left(9x+2\right)\left(x+1\right)$$

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.

$$9x^{2}+11x+2=0$$

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.

$$x=\frac{-11±\sqrt{11^{2}-4\times 9\times 2}}{2\times 9}$$

Square 11.

$$x=\frac{-11±\sqrt{121-4\times 9\times 2}}{2\times 9}$$

Multiply -4 times 9.

$$x=\frac{-11±\sqrt{121-36\times 2}}{2\times 9}$$

Multiply -36 times 2.

$$x=\frac{-11±\sqrt{121-72}}{2\times 9}$$

Add 121 to -72.

$$x=\frac{-11±\sqrt{49}}{2\times 9}$$

Take the square root of 49.

$$x=\frac{-11±7}{2\times 9}$$

Multiply 2 times 9.

$$x=\frac{-11±7}{18}$$

Now solve the equation x=\frac{-11±7}{18} when ± is plus. Add -11 to 7.

$$x=-\frac{4}{18}$$

Reduce the fraction \frac{-4}{18} to lowest terms by extracting and canceling out 2.

$$x=-\frac{2}{9}$$

Now solve the equation x=\frac{-11±7}{18} when ± is minus. Subtract 7 from -11.

$$x=-\frac{18}{18}$$

Divide -18 by 18.

$$x=-1$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{2}{9} for x_{1} and -1 for x_{2}.

$$9x^{2}+11x+2=9\left(x-\left(-\frac{2}{9}\right)\right)\left(x-\left(-1\right)\right)$$

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

$$9x^{2}+11x+2=9\left(x+\frac{2}{9}\right)\left(x+1\right)$$

Add \frac{2}{9} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$9x^{2}+11x+2=9\times \frac{9x+2}{9}\left(x+1\right)$$

Cancel out 9, the greatest common factor in 9 and 9.

$$9x^{2}+11x+2=\left(9x+2\right)\left(x+1\right)$$

Evaluate

$$\left(x+1\right)\left(9x+2\right)$$