Solve 24+2x-12x^2

$$24 + 2 x - 12 x ^ { 2 }$$

Factor

$$2\left(-3x-4\right)\left(2x-3\right)$$

Solution Steps

Factor out 2.

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

Consider 12+x-6x^{2}. Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

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

Factor the expression by grouping. First, the expression needs to be rewritten as -6x^{2}+ax+bx+12. To find a and b, set up a system to be solved.

$$a+b=1 ab=-6\times 12=-72$$

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 -72.

$$-1,72 -2,36 -3,24 -4,18 -6,12 -8,9$$

Calculate the sum for each pair.

$$-1+72=71 -2+36=34 -3+24=21 -4+18=14 -6+12=6 -8+9=1$$

The solution is the pair that gives sum 1.

$$a=9 b=-8$$

Rewrite -6x^{2}+x+12 as \left(-6x^{2}+9x\right)+\left(-8x+12\right).

$$\left(-6x^{2}+9x\right)+\left(-8x+12\right)$$

Factor out -3x in the first and -4 in the second group.

$$-3x\left(2x-3\right)-4\left(2x-3\right)$$

Factor out common term 2x-3 by using distributive property.

$$\left(2x-3\right)\left(-3x-4\right)$$

Rewrite the complete factored expression.

$$2\left(2x-3\right)\left(-3x-4\right)$$

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.

$$-12x^{2}+2x+24=0$$

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.

$$x=\frac{-2±\sqrt{2^{2}-4\left(-12\right)\times 24}}{2\left(-12\right)}$$

Square 2.

$$x=\frac{-2±\sqrt{4-4\left(-12\right)\times 24}}{2\left(-12\right)}$$

Multiply -4 times -12.

$$x=\frac{-2±\sqrt{4+48\times 24}}{2\left(-12\right)}$$

Multiply 48 times 24.

$$x=\frac{-2±\sqrt{4+1152}}{2\left(-12\right)}$$

Add 4 to 1152.

$$x=\frac{-2±\sqrt{1156}}{2\left(-12\right)}$$

Take the square root of 1156.

$$x=\frac{-2±34}{2\left(-12\right)}$$

Multiply 2 times -12.

$$x=\frac{-2±34}{-24}$$

Now solve the equation x=\frac{-2±34}{-24} when ± is plus. Add -2 to 34.

$$x=\frac{32}{-24}$$

Reduce the fraction \frac{32}{-24} to lowest terms by extracting and canceling out 8.

$$x=-\frac{4}{3}$$

Now solve the equation x=\frac{-2±34}{-24} when ± is minus. Subtract 34 from -2.

$$x=-\frac{36}{-24}$$

Reduce the fraction \frac{-36}{-24} to lowest terms by extracting and canceling out 12.

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

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{4}{3} for x_{1} and \frac{3}{2} for x_{2}.

$$-12x^{2}+2x+24=-12\left(x-\left(-\frac{4}{3}\right)\right)\left(x-\frac{3}{2}\right)$$

Simplify all the expressions of the form p-\left(-q\right) to p+q.

$$-12x^{2}+2x+24=-12\left(x+\frac{4}{3}\right)\left(x-\frac{3}{2}\right)$$

Add \frac{4}{3} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$-12x^{2}+2x+24=-12\times \frac{-3x-4}{-3}\left(x-\frac{3}{2}\right)$$

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

$$-12x^{2}+2x+24=-12\times \frac{-3x-4}{-3}\times \frac{-2x+3}{-2}$$

Multiply \frac{-3x-4}{-3} times \frac{-2x+3}{-2} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$-12x^{2}+2x+24=-12\times \frac{\left(-3x-4\right)\left(-2x+3\right)}{-3\left(-2\right)}$$

Multiply -3 times -2.

$$-12x^{2}+2x+24=-12\times \frac{\left(-3x-4\right)\left(-2x+3\right)}{6}$$

Cancel out 6, the greatest common factor in -12 and 6.

$$-12x^{2}+2x+24=-2\left(-3x-4\right)\left(-2x+3\right)$$

Evaluate

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