Solve 2x^2+15x+7

$$2 { x }^{ 2 } +15x+7$$

Factor

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

Solution Steps

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

$$a+b=15 ab=2\times 7=14$$

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 15.

$$a=1 b=14$$

Rewrite 2x^{2}+15x+7 as \left(2x^{2}+x\right)+\left(14x+7\right).

$$\left(2x^{2}+x\right)+\left(14x+7\right)$$

Factor out x in the first and 7 in the second group.

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

Factor out common term 2x+1 by using distributive property.

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

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.

$$2x^{2}+15x+7=0$$

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.

$$x=\frac{-15±\sqrt{15^{2}-4\times 2\times 7}}{2\times 2}$$

Square 15.

$$x=\frac{-15±\sqrt{225-4\times 2\times 7}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-15±\sqrt{225-8\times 7}}{2\times 2}$$

Multiply -8 times 7.

$$x=\frac{-15±\sqrt{225-56}}{2\times 2}$$

Add 225 to -56.

$$x=\frac{-15±\sqrt{169}}{2\times 2}$$

Take the square root of 169.

$$x=\frac{-15±13}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-15±13}{4}$$

Now solve the equation x=\frac{-15±13}{4} when ± is plus. Add -15 to 13.

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

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

$$x=-\frac{1}{2}$$

Now solve the equation x=\frac{-15±13}{4} when ± is minus. Subtract 13 from -15.

$$x=-\frac{28}{4}$$

Divide -28 by 4.

$$x=-7$$

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

$$2x^{2}+15x+7=2\left(x-\left(-\frac{1}{2}\right)\right)\left(x-\left(-7\right)\right)$$

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

$$2x^{2}+15x+7=2\left(x+\frac{1}{2}\right)\left(x+7\right)$$

Add \frac{1}{2} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$2x^{2}+15x+7=2\times \frac{2x+1}{2}\left(x+7\right)$$

Cancel out 2, the greatest common factor in 2 and 2.

$$2x^{2}+15x+7=\left(2x+1\right)\left(x+7\right)$$

Evaluate

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