Solve -24x^2-381x-1422

$$-24 { x }^{ 2 } -381x-1422$$

Factor

$$3\left(-x-6\right)\left(8x+79\right)$$

Solution Steps

Factor out 3.

$$3\left(-8x^{2}-127x-474\right)$$

Consider -8x^{2}-127x-474. Factor the expression by grouping. First, the expression needs to be rewritten as -8x^{2}+ax+bx-474. To find a and b, set up a system to be solved.

$$a+b=-127 ab=-8\left(-474\right)=3792$$

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 3792.

$$-1,-3792 -2,-1896 -3,-1264 -4,-948 -6,-632 -8,-474 -12,-316 -16,-237 -24,-158 -48,-79$$

Calculate the sum for each pair.

$$-1-3792=-3793 -2-1896=-1898 -3-1264=-1267 -4-948=-952 -6-632=-638 -8-474=-482 -12-316=-328 -16-237=-253 -24-158=-182 -48-79=-127$$

The solution is the pair that gives sum -127.

$$a=-48 b=-79$$

Rewrite -8x^{2}-127x-474 as \left(-8x^{2}-48x\right)+\left(-79x-474\right).

$$\left(-8x^{2}-48x\right)+\left(-79x-474\right)$$

Factor out 8x in the first and 79 in the second group.

$$8x\left(-x-6\right)+79\left(-x-6\right)$$

Factor out common term -x-6 by using distributive property.

$$\left(-x-6\right)\left(8x+79\right)$$

Rewrite the complete factored expression.

$$3\left(-x-6\right)\left(8x+79\right)$$

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.

$$-24x^{2}-381x-1422=0$$

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.

$$x=\frac{-\left(-381\right)±\sqrt{\left(-381\right)^{2}-4\left(-24\right)\left(-1422\right)}}{2\left(-24\right)}$$

Square -381.

$$x=\frac{-\left(-381\right)±\sqrt{145161-4\left(-24\right)\left(-1422\right)}}{2\left(-24\right)}$$

Multiply -4 times -24.

$$x=\frac{-\left(-381\right)±\sqrt{145161+96\left(-1422\right)}}{2\left(-24\right)}$$

Multiply 96 times -1422.

$$x=\frac{-\left(-381\right)±\sqrt{145161-136512}}{2\left(-24\right)}$$

Add 145161 to -136512.

$$x=\frac{-\left(-381\right)±\sqrt{8649}}{2\left(-24\right)}$$

Take the square root of 8649.

$$x=\frac{-\left(-381\right)±93}{2\left(-24\right)}$$

The opposite of -381 is 381.

$$x=\frac{381±93}{2\left(-24\right)}$$

Multiply 2 times -24.

$$x=\frac{381±93}{-48}$$

Now solve the equation x=\frac{381±93}{-48} when ± is plus. Add 381 to 93.

$$x=\frac{474}{-48}$$

Reduce the fraction \frac{474}{-48} to lowest terms by extracting and canceling out 6.

$$x=-\frac{79}{8}$$

Now solve the equation x=\frac{381±93}{-48} when ± is minus. Subtract 93 from 381.

$$x=\frac{288}{-48}$$

Divide 288 by -48.

$$x=-6$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{79}{8} for x_{1} and -6 for x_{2}.

$$-24x^{2}-381x-1422=-24\left(x-\left(-\frac{79}{8}\right)\right)\left(x-\left(-6\right)\right)$$

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

$$-24x^{2}-381x-1422=-24\left(x+\frac{79}{8}\right)\left(x+6\right)$$

Add \frac{79}{8} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$-24x^{2}-381x-1422=-24\times \frac{-8x-79}{-8}\left(x+6\right)$$

Cancel out 8, the greatest common factor in -24 and 8.

$$-24x^{2}-381x-1422=3\left(-8x-79\right)\left(x+6\right)$$

Evaluate

$$-24x^{2}-381x-1422$$