Solve -12x^2-13x+14

$$-12 { x }^{ 2 } -13x+14$$

Factor

$$-\left(3x-2\right)\left(4x+7\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as -12x^{2}+ax+bx+14. To find a and b, set up a system to be solved.

$$a+b=-13 ab=-12\times 14=-168$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -168.

$$1,-168 2,-84 3,-56 4,-42 6,-28 7,-24 8,-21 12,-14$$

Calculate the sum for each pair.

$$1-168=-167 2-84=-82 3-56=-53 4-42=-38 6-28=-22 7-24=-17 8-21=-13 12-14=-2$$

The solution is the pair that gives sum -13.

$$a=8 b=-21$$

Rewrite -12x^{2}-13x+14 as \left(-12x^{2}+8x\right)+\left(-21x+14\right).

$$\left(-12x^{2}+8x\right)+\left(-21x+14\right)$$

Factor out -4x in the first and -7 in the second group.

$$-4x\left(3x-2\right)-7\left(3x-2\right)$$

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

$$\left(3x-2\right)\left(-4x-7\right)$$

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.

$$-12x^{2}-13x+14=0$$

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.

$$x=\frac{-\left(-13\right)±\sqrt{\left(-13\right)^{2}-4\left(-12\right)\times 14}}{2\left(-12\right)}$$

Square -13.

$$x=\frac{-\left(-13\right)±\sqrt{169-4\left(-12\right)\times 14}}{2\left(-12\right)}$$

Multiply -4 times -12.

$$x=\frac{-\left(-13\right)±\sqrt{169+48\times 14}}{2\left(-12\right)}$$

Multiply 48 times 14.

$$x=\frac{-\left(-13\right)±\sqrt{169+672}}{2\left(-12\right)}$$

Add 169 to 672.

$$x=\frac{-\left(-13\right)±\sqrt{841}}{2\left(-12\right)}$$

Take the square root of 841.

$$x=\frac{-\left(-13\right)±29}{2\left(-12\right)}$$

The opposite of -13 is 13.

$$x=\frac{13±29}{2\left(-12\right)}$$

Multiply 2 times -12.

$$x=\frac{13±29}{-24}$$

Now solve the equation x=\frac{13±29}{-24} when ± is plus. Add 13 to 29.

$$x=\frac{42}{-24}$$

Reduce the fraction \frac{42}{-24} to lowest terms by extracting and canceling out 6.

$$x=-\frac{7}{4}$$

Now solve the equation x=\frac{13±29}{-24} when ± is minus. Subtract 29 from 13.

$$x=-\frac{16}{-24}$$

Reduce the fraction \frac{-16}{-24} to lowest terms by extracting and canceling out 8.

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

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{7}{4} for x_{1} and \frac{2}{3} for x_{2}.

$$-12x^{2}-13x+14=-12\left(x-\left(-\frac{7}{4}\right)\right)\left(x-\frac{2}{3}\right)$$

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

$$-12x^{2}-13x+14=-12\left(x+\frac{7}{4}\right)\left(x-\frac{2}{3}\right)$$

Add \frac{7}{4} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$-12x^{2}-13x+14=-12\times \frac{-4x-7}{-4}\left(x-\frac{2}{3}\right)$$

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

$$-12x^{2}-13x+14=-12\times \frac{-4x-7}{-4}\times \frac{-3x+2}{-3}$$

Multiply \frac{-4x-7}{-4} times \frac{-3x+2}{-3} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$-12x^{2}-13x+14=-12\times \frac{\left(-4x-7\right)\left(-3x+2\right)}{-4\left(-3\right)}$$

Multiply -4 times -3.

$$-12x^{2}-13x+14=-12\times \frac{\left(-4x-7\right)\left(-3x+2\right)}{12}$$

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

$$-12x^{2}-13x+14=-\left(-4x-7\right)\left(-3x+2\right)$$

Evaluate

$$14-13x-12x^{2}$$