Solve 7x^2+9x+2

$$7 { x }^{ 2 } +9x+2$$

Factor

$$\left(x+1\right)\left(7x+2\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 7x^{2}+ax+bx+2. To find a and b, set up a system to be solved.

$$a+b=9 ab=7\times 2=14$$

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 14.

$$1,14 2,7$$

Calculate the sum for each pair.

$$1+14=15 2+7=9$$

The solution is the pair that gives sum 9.

$$a=2 b=7$$

Rewrite 7x^{2}+9x+2 as \left(7x^{2}+2x\right)+\left(7x+2\right).

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

Factor out x in 7x^{2}+2x.

$$x\left(7x+2\right)+7x+2$$

Factor out common term 7x+2 by using distributive property.

$$\left(7x+2\right)\left(x+1\right)$$

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.

$$7x^{2}+9x+2=0$$

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.

$$x=\frac{-9±\sqrt{9^{2}-4\times 7\times 2}}{2\times 7}$$

Square 9.

$$x=\frac{-9±\sqrt{81-4\times 7\times 2}}{2\times 7}$$

Multiply -4 times 7.

$$x=\frac{-9±\sqrt{81-28\times 2}}{2\times 7}$$

Multiply -28 times 2.

$$x=\frac{-9±\sqrt{81-56}}{2\times 7}$$

Add 81 to -56.

$$x=\frac{-9±\sqrt{25}}{2\times 7}$$

Take the square root of 25.

$$x=\frac{-9±5}{2\times 7}$$

Multiply 2 times 7.

$$x=\frac{-9±5}{14}$$

Now solve the equation x=\frac{-9±5}{14} when ± is plus. Add -9 to 5.

$$x=-\frac{4}{14}$$

Reduce the fraction \frac{-4}{14} to lowest terms by extracting and canceling out 2.

$$x=-\frac{2}{7}$$

Now solve the equation x=\frac{-9±5}{14} when ± is minus. Subtract 5 from -9.

$$x=-\frac{14}{14}$$

Divide -14 by 14.

$$x=-1$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{2}{7} for x_{1} and -1 for x_{2}.

$$7x^{2}+9x+2=7\left(x-\left(-\frac{2}{7}\right)\right)\left(x-\left(-1\right)\right)$$

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

$$7x^{2}+9x+2=7\left(x+\frac{2}{7}\right)\left(x+1\right)$$

Add \frac{2}{7} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$7x^{2}+9x+2=7\times \frac{7x+2}{7}\left(x+1\right)$$

Cancel out 7, the greatest common factor in 7 and 7.

$$7x^{2}+9x+2=\left(7x+2\right)\left(x+1\right)$$

Evaluate

$$\left(x+1\right)\left(7x+2\right)$$