Solve 8x^2+15x+7

$$8 { x }^{ 2 } +15x+7$$

Factor

$$\left(x+1\right)\left(8x+7\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 8x^{2}+ax+bx+7. To find a and b, set up a system to be solved.

$$a+b=15 ab=8\times 7=56$$

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 56.

$$1,56 2,28 4,14 7,8$$

Calculate the sum for each pair.

$$1+56=57 2+28=30 4+14=18 7+8=15$$

The solution is the pair that gives sum 15.

$$a=7 b=8$$

Rewrite 8x^{2}+15x+7 as \left(8x^{2}+7x\right)+\left(8x+7\right).

$$\left(8x^{2}+7x\right)+\left(8x+7\right)$$

Factor out x in 8x^{2}+7x.

$$x\left(8x+7\right)+8x+7$$

Factor out common term 8x+7 by using distributive property.

$$\left(8x+7\right)\left(x+1\right)$$

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.

$$8x^{2}+15x+7=0$$

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.

$$x=\frac{-15±\sqrt{15^{2}-4\times 8\times 7}}{2\times 8}$$

Square 15.

$$x=\frac{-15±\sqrt{225-4\times 8\times 7}}{2\times 8}$$

Multiply -4 times 8.

$$x=\frac{-15±\sqrt{225-32\times 7}}{2\times 8}$$

Multiply -32 times 7.

$$x=\frac{-15±\sqrt{225-224}}{2\times 8}$$

Add 225 to -224.

$$x=\frac{-15±\sqrt{1}}{2\times 8}$$

Take the square root of 1.

$$x=\frac{-15±1}{2\times 8}$$

Multiply 2 times 8.

$$x=\frac{-15±1}{16}$$

Now solve the equation x=\frac{-15±1}{16} when ± is plus. Add -15 to 1.

$$x=-\frac{14}{16}$$

Reduce the fraction \frac{-14}{16} to lowest terms by extracting and canceling out 2.

$$x=-\frac{7}{8}$$

Now solve the equation x=\frac{-15±1}{16} when ± is minus. Subtract 1 from -15.

$$x=-\frac{16}{16}$$

Divide -16 by 16.

$$x=-1$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{7}{8} for x_{1} and -1 for x_{2}.

$$8x^{2}+15x+7=8\left(x-\left(-\frac{7}{8}\right)\right)\left(x-\left(-1\right)\right)$$

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

$$8x^{2}+15x+7=8\left(x+\frac{7}{8}\right)\left(x+1\right)$$

Add \frac{7}{8} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$8x^{2}+15x+7=8\times \frac{8x+7}{8}\left(x+1\right)$$

Cancel out 8, the greatest common factor in 8 and 8.

$$8x^{2}+15x+7=\left(8x+7\right)\left(x+1\right)$$

Evaluate

$$\left(x+1\right)\left(8x+7\right)$$