Solve 2x^2-8x-90

$$2 { x }^{ 2 } -8x-90$$

Factor

$$2\left(x-9\right)\left(x+5\right)$$

Solution Steps

Factor out 2.

$$2\left(x^{2}-4x-45\right)$$

Consider x^{2}-4x-45. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-45. To find a and b, set up a system to be solved.

$$a+b=-4 ab=1\left(-45\right)=-45$$

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 -45.

$$1,-45 3,-15 5,-9$$

Calculate the sum for each pair.

$$1-45=-44 3-15=-12 5-9=-4$$

The solution is the pair that gives sum -4.

$$a=-9 b=5$$

Rewrite x^{2}-4x-45 as \left(x^{2}-9x\right)+\left(5x-45\right).

$$\left(x^{2}-9x\right)+\left(5x-45\right)$$

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

$$x\left(x-9\right)+5\left(x-9\right)$$

Factor out common term x-9 by using distributive property.

$$\left(x-9\right)\left(x+5\right)$$

Rewrite the complete factored expression.

$$2\left(x-9\right)\left(x+5\right)$$

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.

$$2x^{2}-8x-90=0$$

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.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 2\left(-90\right)}}{2\times 2}$$

Square -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\times 2\left(-90\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-8\right)±\sqrt{64-8\left(-90\right)}}{2\times 2}$$

Multiply -8 times -90.

$$x=\frac{-\left(-8\right)±\sqrt{64+720}}{2\times 2}$$

Add 64 to 720.

$$x=\frac{-\left(-8\right)±\sqrt{784}}{2\times 2}$$

Take the square root of 784.

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

The opposite of -8 is 8.

$$x=\frac{8±28}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{8±28}{4}$$

Now solve the equation x=\frac{8±28}{4} when ± is plus. Add 8 to 28.

$$x=\frac{36}{4}$$

Divide 36 by 4.

$$x=9$$

Now solve the equation x=\frac{8±28}{4} when ± is minus. Subtract 28 from 8.

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

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 9 for x_{1} and -5 for x_{2}.

$$2x^{2}-8x-90=2\left(x-9\right)\left(x-\left(-5\right)\right)$$

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

$$2x^{2}-8x-90=2\left(x-9\right)\left(x+5\right)$$

Evaluate

$$2\left(x-9\right)\left(x+5\right)$$