Solve 8x^2=30x+27

$$8 { x }^{ 2 } =30x+27$$

Solve for x

$$x=-\frac{3}{4}=-0.75$$

$$x = \frac{9}{2} = 4\frac{1}{2} = 4.5$$

Steps Using Factoring By Grouping

Subtract 30x from both sides.

$$8x^{2}-30x=27$$

Subtract 27 from both sides.

$$8x^{2}-30x-27=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 8x^{2}+ax+bx-27. To find a and b, set up a system to be solved.

$$a+b=-30 ab=8\left(-27\right)=-216$$

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 -216.

$$1,-216 2,-108 3,-72 4,-54 6,-36 8,-27 9,-24 12,-18$$

Calculate the sum for each pair.

$$1-216=-215 2-108=-106 3-72=-69 4-54=-50 6-36=-30 8-27=-19 9-24=-15 12-18=-6$$

The solution is the pair that gives sum -30.

$$a=-36 b=6$$

Rewrite 8x^{2}-30x-27 as \left(8x^{2}-36x\right)+\left(6x-27\right).

$$\left(8x^{2}-36x\right)+\left(6x-27\right)$$

Factor out 4x in the first and 3 in the second group.

$$4x\left(2x-9\right)+3\left(2x-9\right)$$

Factor out common term 2x-9 by using distributive property.

$$\left(2x-9\right)\left(4x+3\right)$$

To find equation solutions, solve 2x-9=0 and 4x+3=0.

$$x=\frac{9}{2} x=-\frac{3}{4}$$

Steps Using the Quadratic Formula

Subtract 30x from both sides.

$$8x^{2}-30x=27$$

Subtract 27 from both sides.

$$8x^{2}-30x-27=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 8 for a, -30 for b, and -27 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-30\right)±\sqrt{\left(-30\right)^{2}-4\times 8\left(-27\right)}}{2\times 8}$$

Square -30.

$$x=\frac{-\left(-30\right)±\sqrt{900-4\times 8\left(-27\right)}}{2\times 8}$$

Multiply -4 times 8.

$$x=\frac{-\left(-30\right)±\sqrt{900-32\left(-27\right)}}{2\times 8}$$

Multiply -32 times -27.

$$x=\frac{-\left(-30\right)±\sqrt{900+864}}{2\times 8}$$

Add 900 to 864.

$$x=\frac{-\left(-30\right)±\sqrt{1764}}{2\times 8}$$

Take the square root of 1764.

$$x=\frac{-\left(-30\right)±42}{2\times 8}$$

The opposite of -30 is 30.

$$x=\frac{30±42}{2\times 8}$$

Multiply 2 times 8.

$$x=\frac{30±42}{16}$$

Now solve the equation x=\frac{30±42}{16} when ± is plus. Add 30 to 42.

$$x=\frac{72}{16}$$

Reduce the fraction \frac{72}{16} to lowest terms by extracting and canceling out 8.

$$x=\frac{9}{2}$$

Now solve the equation x=\frac{30±42}{16} when ± is minus. Subtract 42 from 30.

$$x=-\frac{12}{16}$$

Reduce the fraction \frac{-12}{16} to lowest terms by extracting and canceling out 4.

$$x=-\frac{3}{4}$$

The equation is now solved.

$$x=\frac{9}{2} x=-\frac{3}{4}$$

Steps for Completing the Square

Subtract 30x from both sides.

$$8x^{2}-30x=27$$

Divide both sides by 8.

$$\frac{8x^{2}-30x}{8}=\frac{27}{8}$$

Dividing by 8 undoes the multiplication by 8.

$$x^{2}+\left(-\frac{30}{8}\right)x=\frac{27}{8}$$

Reduce the fraction \frac{-30}{8} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{15}{4}x=\frac{27}{8}$$

Divide -\frac{15}{4}, the coefficient of the x term, by 2 to get -\frac{15}{8}. Then add the square of -\frac{15}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{15}{4}x+\left(-\frac{15}{8}\right)^{2}=\frac{27}{8}+\left(-\frac{15}{8}\right)^{2}$$

Square -\frac{15}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{15}{4}x+\frac{225}{64}=\frac{27}{8}+\frac{225}{64}$$

Add \frac{27}{8} to \frac{225}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{15}{4}x+\frac{225}{64}=\frac{441}{64}$$

Factor x^{2}-\frac{15}{4}x+\frac{225}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{15}{8}\right)^{2}=\frac{441}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{15}{8}\right)^{2}}=\sqrt{\frac{441}{64}}$$

Simplify.

$$x-\frac{15}{8}=\frac{21}{8} x-\frac{15}{8}=-\frac{21}{8}$$

Add \frac{15}{8} to both sides of the equation.

$$x=\frac{9}{2} x=-\frac{3}{4}$$