Solve =-3x^2-12+12x?

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

Factor

$$-3\left(x-2\right)^{2}$$

Solution Steps

Factor out 3.

$$3\left(-x^{2}-4+4x\right)$$

Consider -x^{2}-4+4x. Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

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

Factor the expression by grouping. First, the expression needs to be rewritten as -x^{2}+ax+bx-4. To find a and b, set up a system to be solved.

$$a+b=4 ab=-\left(-4\right)=4$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. 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 the complete factored expression.

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

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.

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

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.

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

Square 12.

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

Multiply -4 times -3.

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

Multiply 12 times -12.

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

Add 144 to -144.

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

Take the square root of 0.

$$x=\frac{-12±0}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{-12±0}{-6}$$

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 2 for x_{2}.

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

Evaluate

$$-3\left(x-2\right)^{2}$$