Solve 12-3x/2=-3/4x^2+12

$$\frac { 12 - 3 x } { 2 } = - \frac { 3 } { 4 } x ^ { 2 } + 12$$

Solve for x

$$x=-2$$

$$x=4$$

Steps Using Factoring By Grouping

Multiply both sides of the equation by 4, the least common multiple of 2,4.

$$2\left(12-3x\right)=-3x^{2}+48$$

Use the distributive property to multiply 2 by 12-3x.

$$24-6x=-3x^{2}+48$$

Add 3x^{2} to both sides.

$$24-6x+3x^{2}=48$$

Subtract 48 from both sides.

$$24-6x+3x^{2}-48=0$$

Subtract 48 from 24 to get -24.

$$-24-6x+3x^{2}=0$$

Divide both sides by 3.

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

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

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

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

$$a+b=-2 ab=1\left(-8\right)=-8$$

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 -8.

$$1,-8 2,-4$$

Calculate the sum for each pair.

$$1-8=-7 2-4=-2$$

The solution is the pair that gives sum -2.

$$a=-4 b=2$$

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

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

Factor out x in the first and 2 in the second group.

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

Factor out common term x-4 by using distributive property.

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

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

$$x=4 x=-2$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 4, the least common multiple of 2,4.

$$2\left(12-3x\right)=-3x^{2}+48$$

Use the distributive property to multiply 2 by 12-3x.

$$24-6x=-3x^{2}+48$$

Add 3x^{2} to both sides.

$$24-6x+3x^{2}=48$$

Subtract 48 from both sides.

$$24-6x+3x^{2}-48=0$$

Subtract 48 from 24 to get -24.

$$-24-6x+3x^{2}=0$$

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.

$$3x^{2}-6x-24=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -6 for b, and -24 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 3\left(-24\right)}}{2\times 3}$$

Square -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\times 3\left(-24\right)}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-\left(-6\right)±\sqrt{36-12\left(-24\right)}}{2\times 3}$$

Multiply -12 times -24.

$$x=\frac{-\left(-6\right)±\sqrt{36+288}}{2\times 3}$$

Add 36 to 288.

$$x=\frac{-\left(-6\right)±\sqrt{324}}{2\times 3}$$

Take the square root of 324.

$$x=\frac{-\left(-6\right)±18}{2\times 3}$$

The opposite of -6 is 6.

$$x=\frac{6±18}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{6±18}{6}$$

Now solve the equation x=\frac{6±18}{6} when ± is plus. Add 6 to 18.

$$x=\frac{24}{6}$$

Divide 24 by 6.

$$x=4$$

Now solve the equation x=\frac{6±18}{6} when ± is minus. Subtract 18 from 6.

$$x=-\frac{12}{6}$$

Divide -12 by 6.

$$x=-2$$

The equation is now solved.

$$x=4 x=-2$$

Steps for Completing the Square

Multiply both sides of the equation by 4, the least common multiple of 2,4.

$$2\left(12-3x\right)=-3x^{2}+48$$

Use the distributive property to multiply 2 by 12-3x.

$$24-6x=-3x^{2}+48$$

Add 3x^{2} to both sides.

$$24-6x+3x^{2}=48$$

Subtract 24 from both sides.

$$-6x+3x^{2}=48-24$$

Subtract 24 from 48 to get 24.

$$-6x+3x^{2}=24$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$3x^{2}-6x=24$$

Divide both sides by 3.

$$\frac{3x^{2}-6x}{3}=\frac{24}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$x^{2}+\left(-\frac{6}{3}\right)x=\frac{24}{3}$$

Divide -6 by 3.

$$x^{2}-2x=\frac{24}{3}$$

Divide 24 by 3.

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

Divide -2, the coefficient of the x term, by 2 to get -1. Then add the square of -1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-2x+1=8+1$$

Add 8 to 1.

$$x^{2}-2x+1=9$$

Factor x^{2}-2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-1\right)^{2}=9$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{9}$$

Simplify.

$$x-1=3 x-1=-3$$

Add 1 to both sides of the equation.

$$x=4 x=-2$$