Solve -6k^2-13k-6

$$-6 { k }^{ 2 } -13k-6$$

Factor

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

Solution Steps

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

$$a+b=-13 ab=-6\left(-6\right)=36$$

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 36.

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

Calculate the sum for each pair.

$$-1-36=-37 -2-18=-20 -3-12=-15 -4-9=-13 -6-6=-12$$

The solution is the pair that gives sum -13.

$$a=-4 b=-9$$

Rewrite -6k^{2}-13k-6 as \left(-6k^{2}-4k\right)+\left(-9k-6\right).

$$\left(-6k^{2}-4k\right)+\left(-9k-6\right)$$

Factor out 2k in the first and 3 in the second group.

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

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

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

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.

$$-6k^{2}-13k-6=0$$

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.

$$k=\frac{-\left(-13\right)±\sqrt{\left(-13\right)^{2}-4\left(-6\right)\left(-6\right)}}{2\left(-6\right)}$$

Square -13.

$$k=\frac{-\left(-13\right)±\sqrt{169-4\left(-6\right)\left(-6\right)}}{2\left(-6\right)}$$

Multiply -4 times -6.

$$k=\frac{-\left(-13\right)±\sqrt{169+24\left(-6\right)}}{2\left(-6\right)}$$

Multiply 24 times -6.

$$k=\frac{-\left(-13\right)±\sqrt{169-144}}{2\left(-6\right)}$$

Add 169 to -144.

$$k=\frac{-\left(-13\right)±\sqrt{25}}{2\left(-6\right)}$$

Take the square root of 25.

$$k=\frac{-\left(-13\right)±5}{2\left(-6\right)}$$

The opposite of -13 is 13.

$$k=\frac{13±5}{2\left(-6\right)}$$

Multiply 2 times -6.

$$k=\frac{13±5}{-12}$$

Now solve the equation k=\frac{13±5}{-12} when ± is plus. Add 13 to 5.

$$k=\frac{18}{-12}$$

Reduce the fraction \frac{18}{-12} to lowest terms by extracting and canceling out 6.

$$k=-\frac{3}{2}$$

Now solve the equation k=\frac{13±5}{-12} when ± is minus. Subtract 5 from 13.

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

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

$$k=-\frac{2}{3}$$

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

$$-6k^{2}-13k-6=-6\left(k-\left(-\frac{3}{2}\right)\right)\left(k-\left(-\frac{2}{3}\right)\right)$$

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

$$-6k^{2}-13k-6=-6\left(k+\frac{3}{2}\right)\left(k+\frac{2}{3}\right)$$

Add \frac{3}{2} to k by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$-6k^{2}-13k-6=-6\times \frac{-2k-3}{-2}\left(k+\frac{2}{3}\right)$$

Add \frac{2}{3} to k by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$-6k^{2}-13k-6=-6\times \frac{-2k-3}{-2}\times \frac{-3k-2}{-3}$$

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

$$-6k^{2}-13k-6=-6\times \frac{\left(-2k-3\right)\left(-3k-2\right)}{-2\left(-3\right)}$$

Multiply -2 times -3.

$$-6k^{2}-13k-6=-6\times \frac{\left(-2k-3\right)\left(-3k-2\right)}{6}$$

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

$$-6k^{2}-13k-6=-\left(-2k-3\right)\left(-3k-2\right)$$

Evaluate

$$-6k^{2}-13k-6$$