Solve 5-6x-8x^2

$$5 - 6 x - 8 x ^ { 2 }$$

Factor

$$-\left(2x-1\right)\left(4x+5\right)$$

Solution Steps

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-8x^{2}-6x+5$$

Factor the expression by grouping. First, the expression needs to be rewritten as -8x^{2}+ax+bx+5. To find a and b, set up a system to be solved.

$$a+b=-6 ab=-8\times 5=-40$$

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 -40.

$$1,-40 2,-20 4,-10 5,-8$$

Calculate the sum for each pair.

$$1-40=-39 2-20=-18 4-10=-6 5-8=-3$$

The solution is the pair that gives sum -6.

$$a=4 b=-10$$

Rewrite -8x^{2}-6x+5 as \left(-8x^{2}+4x\right)+\left(-10x+5\right).

$$\left(-8x^{2}+4x\right)+\left(-10x+5\right)$$

Factor out -4x in the first and -5 in the second group.

$$-4x\left(2x-1\right)-5\left(2x-1\right)$$

Factor out common term 2x-1 by using distributive property.

$$\left(2x-1\right)\left(-4x-5\right)$$

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.

$$-8x^{2}-6x+5=0$$

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.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\left(-8\right)\times 5}}{2\left(-8\right)}$$

Square -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\left(-8\right)\times 5}}{2\left(-8\right)}$$

Multiply -4 times -8.

$$x=\frac{-\left(-6\right)±\sqrt{36+32\times 5}}{2\left(-8\right)}$$

Multiply 32 times 5.

$$x=\frac{-\left(-6\right)±\sqrt{36+160}}{2\left(-8\right)}$$

Add 36 to 160.

$$x=\frac{-\left(-6\right)±\sqrt{196}}{2\left(-8\right)}$$

Take the square root of 196.

$$x=\frac{-\left(-6\right)±14}{2\left(-8\right)}$$

The opposite of -6 is 6.

$$x=\frac{6±14}{2\left(-8\right)}$$

Multiply 2 times -8.

$$x=\frac{6±14}{-16}$$

Now solve the equation x=\frac{6±14}{-16} when ± is plus. Add 6 to 14.

$$x=\frac{20}{-16}$$

Reduce the fraction \frac{20}{-16} to lowest terms by extracting and canceling out 4.

$$x=-\frac{5}{4}$$

Now solve the equation x=\frac{6±14}{-16} when ± is minus. Subtract 14 from 6.

$$x=-\frac{8}{-16}$$

Reduce the fraction \frac{-8}{-16} to lowest terms by extracting and canceling out 8.

$$x=\frac{1}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{5}{4} for x_{1} and \frac{1}{2} for x_{2}.

$$-8x^{2}-6x+5=-8\left(x-\left(-\frac{5}{4}\right)\right)\left(x-\frac{1}{2}\right)$$

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

$$-8x^{2}-6x+5=-8\left(x+\frac{5}{4}\right)\left(x-\frac{1}{2}\right)$$

Add \frac{5}{4} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$-8x^{2}-6x+5=-8\times \frac{-4x-5}{-4}\left(x-\frac{1}{2}\right)$$

Subtract \frac{1}{2} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$-8x^{2}-6x+5=-8\times \frac{-4x-5}{-4}\times \frac{-2x+1}{-2}$$

Multiply \frac{-4x-5}{-4} times \frac{-2x+1}{-2} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$-8x^{2}-6x+5=-8\times \frac{\left(-4x-5\right)\left(-2x+1\right)}{-4\left(-2\right)}$$

Multiply -4 times -2.

$$-8x^{2}-6x+5=-8\times \frac{\left(-4x-5\right)\left(-2x+1\right)}{8}$$

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

$$-8x^{2}-6x+5=-\left(-4x-5\right)\left(-2x+1\right)$$

Evaluate

$$5-6x-8x^{2}$$