Solve 8x^2-5x-22=

$$8 { x }^{ 2 } -5x-22=$$

Factor

$$\left(x-2\right)\left(8x+11\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 8x^{2}+ax+bx-22. To find a and b, set up a system to be solved.

$$a+b=-5 ab=8\left(-22\right)=-176$$

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 -176.

$$1,-176 2,-88 4,-44 8,-22 11,-16$$

Calculate the sum for each pair.

$$1-176=-175 2-88=-86 4-44=-40 8-22=-14 11-16=-5$$

The solution is the pair that gives sum -5.

$$a=-16 b=11$$

Rewrite 8x^{2}-5x-22 as \left(8x^{2}-16x\right)+\left(11x-22\right).

$$\left(8x^{2}-16x\right)+\left(11x-22\right)$$

Factor out 8x in the first and 11 in the second group.

$$8x\left(x-2\right)+11\left(x-2\right)$$

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

$$\left(x-2\right)\left(8x+11\right)$$

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.

$$8x^{2}-5x-22=0$$

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.

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 8\left(-22\right)}}{2\times 8}$$

Square -5.

$$x=\frac{-\left(-5\right)±\sqrt{25-4\times 8\left(-22\right)}}{2\times 8}$$

Multiply -4 times 8.

$$x=\frac{-\left(-5\right)±\sqrt{25-32\left(-22\right)}}{2\times 8}$$

Multiply -32 times -22.

$$x=\frac{-\left(-5\right)±\sqrt{25+704}}{2\times 8}$$

Add 25 to 704.

$$x=\frac{-\left(-5\right)±\sqrt{729}}{2\times 8}$$

Take the square root of 729.

$$x=\frac{-\left(-5\right)±27}{2\times 8}$$

The opposite of -5 is 5.

$$x=\frac{5±27}{2\times 8}$$

Multiply 2 times 8.

$$x=\frac{5±27}{16}$$

Now solve the equation x=\frac{5±27}{16} when ± is plus. Add 5 to 27.

$$x=\frac{32}{16}$$

Divide 32 by 16.

$$x=2$$

Now solve the equation x=\frac{5±27}{16} when ± is minus. Subtract 27 from 5.

$$x=-\frac{22}{16}$$

Reduce the fraction \frac{-22}{16} to lowest terms by extracting and canceling out 2.

$$x=-\frac{11}{8}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 2 for x_{1} and -\frac{11}{8} for x_{2}.

$$8x^{2}-5x-22=8\left(x-2\right)\left(x-\left(-\frac{11}{8}\right)\right)$$

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

$$8x^{2}-5x-22=8\left(x-2\right)\left(x+\frac{11}{8}\right)$$

Add \frac{11}{8} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$8x^{2}-5x-22=8\left(x-2\right)\times \frac{8x+11}{8}$$

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

$$8x^{2}-5x-22=\left(x-2\right)\left(8x+11\right)$$

Evaluate

$$\left(x-2\right)\left(8x+11\right)$$