Solve x^2+5x/3-x+3/2=2x^2-2/8

$$\frac { x ^ { 2 } + 5 x } { 3 } - \frac { x + 3 } { 2 } = \frac { 2 x ^ { 2 } - 2 } { 8 }$$

Solve for x

$$x=-15$$

$$x=1$$

Steps Using Factoring By Grouping

Multiply both sides of the equation by 24, the least common multiple of 3,2,8.

$$8\left(x^{2}+5x\right)-12\left(x+3\right)=3\left(2x^{2}-2\right)$$

Use the distributive property to multiply 8 by x^{2}+5x.

$$8x^{2}+40x-12\left(x+3\right)=3\left(2x^{2}-2\right)$$

Use the distributive property to multiply -12 by x+3.

$$8x^{2}+40x-12x-36=3\left(2x^{2}-2\right)$$

Combine 40x and -12x to get 28x.

$$8x^{2}+28x-36=3\left(2x^{2}-2\right)$$

Use the distributive property to multiply 3 by 2x^{2}-2.

$$8x^{2}+28x-36=6x^{2}-6$$

Subtract 6x^{2} from both sides.

$$8x^{2}+28x-36-6x^{2}=-6$$

Combine 8x^{2} and -6x^{2} to get 2x^{2}.

$$2x^{2}+28x-36=-6$$

Add 6 to both sides.

$$2x^{2}+28x-36+6=0$$

Add -36 and 6 to get -30.

$$2x^{2}+28x-30=0$$

Divide both sides by 2.

$$x^{2}+14x-15=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-15. To find a and b, set up a system to be solved.

$$a+b=14 ab=1\left(-15\right)=-15$$

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 -15.

$$-1,15 -3,5$$

Calculate the sum for each pair.

$$-1+15=14 -3+5=2$$

The solution is the pair that gives sum 14.

$$a=-1 b=15$$

Rewrite x^{2}+14x-15 as \left(x^{2}-x\right)+\left(15x-15\right).

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

Factor out x in the first and 15 in the second group.

$$x\left(x-1\right)+15\left(x-1\right)$$

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

$$\left(x-1\right)\left(x+15\right)$$

To find equation solutions, solve x-1=0 and x+15=0.

$$x=1 x=-15$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 24, the least common multiple of 3,2,8.

$$8\left(x^{2}+5x\right)-12\left(x+3\right)=3\left(2x^{2}-2\right)$$

Use the distributive property to multiply 8 by x^{2}+5x.

$$8x^{2}+40x-12\left(x+3\right)=3\left(2x^{2}-2\right)$$

Use the distributive property to multiply -12 by x+3.

$$8x^{2}+40x-12x-36=3\left(2x^{2}-2\right)$$

Combine 40x and -12x to get 28x.

$$8x^{2}+28x-36=3\left(2x^{2}-2\right)$$

Use the distributive property to multiply 3 by 2x^{2}-2.

$$8x^{2}+28x-36=6x^{2}-6$$

Subtract 6x^{2} from both sides.

$$8x^{2}+28x-36-6x^{2}=-6$$

Combine 8x^{2} and -6x^{2} to get 2x^{2}.

$$2x^{2}+28x-36=-6$$

Add 6 to both sides.

$$2x^{2}+28x-36+6=0$$

Add -36 and 6 to get -30.

$$2x^{2}+28x-30=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 28 for b, and -30 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-28±\sqrt{28^{2}-4\times 2\left(-30\right)}}{2\times 2}$$

Square 28.

$$x=\frac{-28±\sqrt{784-4\times 2\left(-30\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-28±\sqrt{784-8\left(-30\right)}}{2\times 2}$$

Multiply -8 times -30.

$$x=\frac{-28±\sqrt{784+240}}{2\times 2}$$

Add 784 to 240.

$$x=\frac{-28±\sqrt{1024}}{2\times 2}$$

Take the square root of 1024.

$$x=\frac{-28±32}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-28±32}{4}$$

Now solve the equation x=\frac{-28±32}{4} when ± is plus. Add -28 to 32.

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

Divide 4 by 4.

$$x=1$$

Now solve the equation x=\frac{-28±32}{4} when ± is minus. Subtract 32 from -28.

$$x=-\frac{60}{4}$$

Divide -60 by 4.

$$x=-15$$

The equation is now solved.

$$x=1 x=-15$$

Steps for Completing the Square

Multiply both sides of the equation by 24, the least common multiple of 3,2,8.

$$8\left(x^{2}+5x\right)-12\left(x+3\right)=3\left(2x^{2}-2\right)$$

Use the distributive property to multiply 8 by x^{2}+5x.

$$8x^{2}+40x-12\left(x+3\right)=3\left(2x^{2}-2\right)$$

Use the distributive property to multiply -12 by x+3.

$$8x^{2}+40x-12x-36=3\left(2x^{2}-2\right)$$

Combine 40x and -12x to get 28x.

$$8x^{2}+28x-36=3\left(2x^{2}-2\right)$$

Use the distributive property to multiply 3 by 2x^{2}-2.

$$8x^{2}+28x-36=6x^{2}-6$$

Subtract 6x^{2} from both sides.

$$8x^{2}+28x-36-6x^{2}=-6$$

Combine 8x^{2} and -6x^{2} to get 2x^{2}.

$$2x^{2}+28x-36=-6$$

Add 36 to both sides.

$$2x^{2}+28x=-6+36$$

Add -6 and 36 to get 30.

$$2x^{2}+28x=30$$

Divide both sides by 2.

$$\frac{2x^{2}+28x}{2}=\frac{30}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{28}{2}x=\frac{30}{2}$$

Divide 28 by 2.

$$x^{2}+14x=\frac{30}{2}$$

Divide 30 by 2.

$$x^{2}+14x=15$$

Divide 14, the coefficient of the x term, by 2 to get 7. Then add the square of 7 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+14x+7^{2}=15+7^{2}$$

Square 7.

$$x^{2}+14x+49=15+49$$

Add 15 to 49.

$$x^{2}+14x+49=64$$

Factor x^{2}+14x+49. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+7\right)^{2}=64$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+7\right)^{2}}=\sqrt{64}$$

Simplify.

$$x+7=8 x+7=-8$$

Subtract 7 from both sides of the equation.

$$x=1 x=-15$$