Solve 3x^2=12(x-1)

$$3 x ^ { 2 } = 12 ( x - 1 )$$

Solve for x

$$x=2$$

Steps Using Factoring By Grouping

Use the distributive property to multiply 12 by x-1.

$$3x^{2}=12x-12$$

Subtract 12x from both sides.

$$3x^{2}-12x=-12$$

Add 12 to both sides.

$$3x^{2}-12x+12=0$$

Divide both sides by 3.

$$x^{2}-4x+4=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+4. To find a and b, set up a system to be solved.

$$a+b=-4 ab=1\times 4=4$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 4.

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

Calculate the sum for each pair.

$$-1-4=-5 -2-2=-4$$

The solution is the pair that gives sum -4.

$$a=-2 b=-2$$

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

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

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

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

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

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

Rewrite as a binomial square.

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

To find equation solution, solve x-2=0.

$$x=2$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 12 by x-1.

$$3x^{2}=12x-12$$

Subtract 12x from both sides.

$$3x^{2}-12x=-12$$

Add 12 to both sides.

$$3x^{2}-12x+12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -12 for b, and 12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 3\times 12}}{2\times 3}$$

Square -12.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\times 3\times 12}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-\left(-12\right)±\sqrt{144-12\times 12}}{2\times 3}$$

Multiply -12 times 12.

$$x=\frac{-\left(-12\right)±\sqrt{144-144}}{2\times 3}$$

Add 144 to -144.

$$x=\frac{-\left(-12\right)±\sqrt{0}}{2\times 3}$$

Take the square root of 0.

$$x=-\frac{-12}{2\times 3}$$

The opposite of -12 is 12.

$$x=\frac{12}{2\times 3}$$

Multiply 2 times 3.

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

Divide 12 by 6.

$$x=2$$

Steps for Completing the Square

Use the distributive property to multiply 12 by x-1.

$$3x^{2}=12x-12$$

Subtract 12x from both sides.

$$3x^{2}-12x=-12$$

Divide both sides by 3.

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

Dividing by 3 undoes the multiplication by 3.

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

Divide -12 by 3.

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

Divide -12 by 3.

$$x^{2}-4x=-4$$

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

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

Square -2.

$$x^{2}-4x+4=-4+4$$

Add -4 to 4.

$$x^{2}-4x+4=0$$

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

$$\left(x-2\right)^{2}=0$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{0}$$

Simplify.

$$x-2=0 x-2=0$$

Add 2 to both sides of the equation.

$$x=2 x=2$$

The equation is now solved. Solutions are the same.

$$x=2$$