Solve 2x^2-7x-15

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

Factor

$$\left(x-5\right)\left(2x+3\right)$$

Solution Steps

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

$$a+b=-7 ab=2\left(-15\right)=-30$$

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 -30.

$$1,-30 2,-15 3,-10 5,-6$$

Calculate the sum for each pair.

$$1-30=-29 2-15=-13 3-10=-7 5-6=-1$$

The solution is the pair that gives sum -7.

$$a=-10 b=3$$

Rewrite 2x^{2}-7x-15 as \left(2x^{2}-10x\right)+\left(3x-15\right).

$$\left(2x^{2}-10x\right)+\left(3x-15\right)$$

Factor out 2x in the first and 3 in the second group.

$$2x\left(x-5\right)+3\left(x-5\right)$$

Factor out common term x-5 by using distributive property.

$$\left(x-5\right)\left(2x+3\right)$$

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.

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

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.

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

Square -7.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\times 2\left(-15\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-7\right)±\sqrt{49-8\left(-15\right)}}{2\times 2}$$

Multiply -8 times -15.

$$x=\frac{-\left(-7\right)±\sqrt{49+120}}{2\times 2}$$

Add 49 to 120.

$$x=\frac{-\left(-7\right)±\sqrt{169}}{2\times 2}$$

Take the square root of 169.

$$x=\frac{-\left(-7\right)±13}{2\times 2}$$

The opposite of -7 is 7.

$$x=\frac{7±13}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{7±13}{4}$$

Now solve the equation x=\frac{7±13}{4} when ± is plus. Add 7 to 13.

$$x=\frac{20}{4}$$

Divide 20 by 4.

$$x=5$$

Now solve the equation x=\frac{7±13}{4} when ± is minus. Subtract 13 from 7.

$$x=-\frac{6}{4}$$

Reduce the fraction \frac{-6}{4} to lowest terms by extracting and canceling out 2.

$$x=-\frac{3}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 5 for x_{1} and -\frac{3}{2} for x_{2}.

$$2x^{2}-7x-15=2\left(x-5\right)\left(x-\left(-\frac{3}{2}\right)\right)$$

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

$$2x^{2}-7x-15=2\left(x-5\right)\left(x+\frac{3}{2}\right)$$

Add \frac{3}{2} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$2x^{2}-7x-15=2\left(x-5\right)\times \frac{2x+3}{2}$$

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

$$2x^{2}-7x-15=\left(x-5\right)\left(2x+3\right)$$

Evaluate

$$\left(x-5\right)\left(2x+3\right)$$