Solve -3x^2+38x-24

$$-3 { x }^{ 2 } +38x-24$$

Factor

$$\left(12-x\right)\left(3x-2\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as -3x^{2}+ax+bx-24. To find a and b, set up a system to be solved.

$$a+b=38 ab=-3\left(-24\right)=72$$

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 72.

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

Calculate the sum for each pair.

$$1+72=73 2+36=38 3+24=27 4+18=22 6+12=18 8+9=17$$

The solution is the pair that gives sum 38.

$$a=36 b=2$$

Rewrite -3x^{2}+38x-24 as \left(-3x^{2}+36x\right)+\left(2x-24\right).

$$\left(-3x^{2}+36x\right)+\left(2x-24\right)$$

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

$$3x\left(-x+12\right)-2\left(-x+12\right)$$

Factor out common term -x+12 by using distributive property.

$$\left(-x+12\right)\left(3x-2\right)$$

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.

$$-3x^{2}+38x-24=0$$

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.

$$x=\frac{-38±\sqrt{38^{2}-4\left(-3\right)\left(-24\right)}}{2\left(-3\right)}$$

Square 38.

$$x=\frac{-38±\sqrt{1444-4\left(-3\right)\left(-24\right)}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-38±\sqrt{1444+12\left(-24\right)}}{2\left(-3\right)}$$

Multiply 12 times -24.

$$x=\frac{-38±\sqrt{1444-288}}{2\left(-3\right)}$$

Add 1444 to -288.

$$x=\frac{-38±\sqrt{1156}}{2\left(-3\right)}$$

Take the square root of 1156.

$$x=\frac{-38±34}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{-38±34}{-6}$$

Now solve the equation x=\frac{-38±34}{-6} when ± is plus. Add -38 to 34.

$$x=-\frac{4}{-6}$$

Reduce the fraction \frac{-4}{-6} to lowest terms by extracting and canceling out 2.

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

Now solve the equation x=\frac{-38±34}{-6} when ± is minus. Subtract 34 from -38.

$$x=-\frac{72}{-6}$$

Divide -72 by -6.

$$x=12$$

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 12 for x_{2}.

$$-3x^{2}+38x-24=-3\left(x-\frac{2}{3}\right)\left(x-12\right)$$

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

$$-3x^{2}+38x-24=-3\times \frac{-3x+2}{-3}\left(x-12\right)$$

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

$$-3x^{2}+38x-24=\left(-3x+2\right)\left(x-12\right)$$

Evaluate

$$\left(12-x\right)\left(3x-2\right)$$