Solve 2x^2+7x-15

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

Factor

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

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 positive, the positive number has greater absolute value than the negative. 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.

$$a=-3 b=10$$

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

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

Factor out x in the first and 5 in the second group.

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

Factor out common term 2x-3 by using distributive property.

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

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.

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

Square 7.

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

Multiply -4 times 2.

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

Multiply -8 times -15.

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

Add 49 to 120.

$$x=\frac{-7±\sqrt{169}}{2\times 2}$$

Take the square root of 169.

$$x=\frac{-7±13}{2\times 2}$$

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.

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

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

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

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

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

Divide -20 by 4.

$$x=-5$$

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

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

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

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

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

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

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

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

Evaluate

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