Solve 22x-12-6x^2

$$22 x - 12 - 6 x ^ { 2 }$$

Factor

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

Solution Steps

Factor out 2.

$$2\left(11x-6-3x^{2}\right)$$

Consider 11x-6-3x^{2}. Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-3x^{2}+11x-6$$

Factor the expression by grouping. First, the expression needs to be rewritten as -3x^{2}+ax+bx-6. To find a and b, set up a system to be solved.

$$a+b=11 ab=-3\left(-6\right)=18$$

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 18.

$$1,18 2,9 3,6$$

Calculate the sum for each pair.

$$1+18=19 2+9=11 3+6=9$$

The solution is the pair that gives sum 11.

$$a=9 b=2$$

Rewrite -3x^{2}+11x-6 as \left(-3x^{2}+9x\right)+\left(2x-6\right).

$$\left(-3x^{2}+9x\right)+\left(2x-6\right)$$

Factor out 3x in the first and -2 in the second group.

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

Factor out common term -x+3 by using distributive property.

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

Rewrite the complete factored expression.

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

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.

$$-6x^{2}+22x-12=0$$

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.

$$x=\frac{-22±\sqrt{22^{2}-4\left(-6\right)\left(-12\right)}}{2\left(-6\right)}$$

Square 22.

$$x=\frac{-22±\sqrt{484-4\left(-6\right)\left(-12\right)}}{2\left(-6\right)}$$

Multiply -4 times -6.

$$x=\frac{-22±\sqrt{484+24\left(-12\right)}}{2\left(-6\right)}$$

Multiply 24 times -12.

$$x=\frac{-22±\sqrt{484-288}}{2\left(-6\right)}$$

Add 484 to -288.

$$x=\frac{-22±\sqrt{196}}{2\left(-6\right)}$$

Take the square root of 196.

$$x=\frac{-22±14}{2\left(-6\right)}$$

Multiply 2 times -6.

$$x=\frac{-22±14}{-12}$$

Now solve the equation x=\frac{-22±14}{-12} when ± is plus. Add -22 to 14.

$$x=-\frac{8}{-12}$$

Reduce the fraction \frac{-8}{-12} to lowest terms by extracting and canceling out 4.

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

Now solve the equation x=\frac{-22±14}{-12} when ± is minus. Subtract 14 from -22.

$$x=-\frac{36}{-12}$$

Divide -36 by -12.

$$x=3$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{2}{3} for x_{1} and 3 for x_{2}.

$$-6x^{2}+22x-12=-6\left(x-\frac{2}{3}\right)\left(x-3\right)$$

Subtract \frac{2}{3} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$-6x^{2}+22x-12=-6\times \frac{-3x+2}{-3}\left(x-3\right)$$

Cancel out 3, the greatest common factor in -6 and 3.

$$-6x^{2}+22x-12=2\left(-3x+2\right)\left(x-3\right)$$

Evaluate

$$-6x^{2}+22x-12$$