Solve 7x^2-8x-15=

$$7 { x }^{ 2 } -8x-15=$$

Factor

$$\left(7x-15\right)\left(x+1\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 7x^{2}+ax+bx-15. To find a and b, set up a system to be solved.

$$a+b=-8 ab=7\left(-15\right)=-105$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -105.

$$1,-105 3,-35 5,-21 7,-15$$

Calculate the sum for each pair.

$$1-105=-104 3-35=-32 5-21=-16 7-15=-8$$

The solution is the pair that gives sum -8.

$$a=-15 b=7$$

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

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

Factor out x in 7x^{2}-15x.

$$x\left(7x-15\right)+7x-15$$

Factor out common term 7x-15 by using distributive property.

$$\left(7x-15\right)\left(x+1\right)$$

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.

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

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.

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

Square -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\times 7\left(-15\right)}}{2\times 7}$$

Multiply -4 times 7.

$$x=\frac{-\left(-8\right)±\sqrt{64-28\left(-15\right)}}{2\times 7}$$

Multiply -28 times -15.

$$x=\frac{-\left(-8\right)±\sqrt{64+420}}{2\times 7}$$

Add 64 to 420.

$$x=\frac{-\left(-8\right)±\sqrt{484}}{2\times 7}$$

Take the square root of 484.

$$x=\frac{-\left(-8\right)±22}{2\times 7}$$

The opposite of -8 is 8.

$$x=\frac{8±22}{2\times 7}$$

Multiply 2 times 7.

$$x=\frac{8±22}{14}$$

Now solve the equation x=\frac{8±22}{14} when ± is plus. Add 8 to 22.

$$x=\frac{30}{14}$$

Reduce the fraction \frac{30}{14} to lowest terms by extracting and canceling out 2.

$$x=\frac{15}{7}$$

Now solve the equation x=\frac{8±22}{14} when ± is minus. Subtract 22 from 8.

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

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{15}{7} for x_{1} and -1 for x_{2}.

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

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

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

Subtract \frac{15}{7} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

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

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

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

Evaluate

$$\left(7x-15\right)\left(x+1\right)$$