Solve 6k^2-24k-192

$$6 { k }^{ 2 } -24k-192$$

Factor

$$6\left(k-8\right)\left(k+4\right)$$

Solution Steps

Factor out 6.

$$6\left(k^{2}-4k-32\right)$$

Consider k^{2}-4k-32. Factor the expression by grouping. First, the expression needs to be rewritten as k^{2}+ak+bk-32. To find a and b, set up a system to be solved.

$$a+b=-4 ab=1\left(-32\right)=-32$$

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 -32.

$$1,-32 2,-16 4,-8$$

Calculate the sum for each pair.

$$1-32=-31 2-16=-14 4-8=-4$$

The solution is the pair that gives sum -4.

$$a=-8 b=4$$

Rewrite k^{2}-4k-32 as \left(k^{2}-8k\right)+\left(4k-32\right).

$$\left(k^{2}-8k\right)+\left(4k-32\right)$$

Factor out k in the first and 4 in the second group.

$$k\left(k-8\right)+4\left(k-8\right)$$

Factor out common term k-8 by using distributive property.

$$\left(k-8\right)\left(k+4\right)$$

Rewrite the complete factored expression.

$$6\left(k-8\right)\left(k+4\right)$$

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.

$$6k^{2}-24k-192=0$$

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.

$$k=\frac{-\left(-24\right)±\sqrt{\left(-24\right)^{2}-4\times 6\left(-192\right)}}{2\times 6}$$

Square -24.

$$k=\frac{-\left(-24\right)±\sqrt{576-4\times 6\left(-192\right)}}{2\times 6}$$

Multiply -4 times 6.

$$k=\frac{-\left(-24\right)±\sqrt{576-24\left(-192\right)}}{2\times 6}$$

Multiply -24 times -192.

$$k=\frac{-\left(-24\right)±\sqrt{576+4608}}{2\times 6}$$

Add 576 to 4608.

$$k=\frac{-\left(-24\right)±\sqrt{5184}}{2\times 6}$$

Take the square root of 5184.

$$k=\frac{-\left(-24\right)±72}{2\times 6}$$

The opposite of -24 is 24.

$$k=\frac{24±72}{2\times 6}$$

Multiply 2 times 6.

$$k=\frac{24±72}{12}$$

Now solve the equation k=\frac{24±72}{12} when ± is plus. Add 24 to 72.

$$k=\frac{96}{12}$$

Divide 96 by 12.

$$k=8$$

Now solve the equation k=\frac{24±72}{12} when ± is minus. Subtract 72 from 24.

$$k=-\frac{48}{12}$$

Divide -48 by 12.

$$k=-4$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 8 for x_{1} and -4 for x_{2}.

$$6k^{2}-24k-192=6\left(k-8\right)\left(k-\left(-4\right)\right)$$

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

$$6k^{2}-24k-192=6\left(k-8\right)\left(k+4\right)$$

Evaluate

$$6\left(k-8\right)\left(k+4\right)$$