Solve -12x^2-48x-36

$$-12 { x }^{ 2 } -48x-36$$

Factor

$$-12\left(x+1\right)\left(x+3\right)$$

Solution Steps

Factor out 12.

$$12\left(-x^{2}-4x-3\right)$$

Consider -x^{2}-4x-3. Factor the expression by grouping. First, the expression needs to be rewritten as -x^{2}+ax+bx-3. To find a and b, set up a system to be solved.

$$a+b=-4 ab=-\left(-3\right)=3$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. The only such pair is the system solution.

$$a=-1 b=-3$$

Rewrite -x^{2}-4x-3 as \left(-x^{2}-x\right)+\left(-3x-3\right).

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

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

$$x\left(-x-1\right)+3\left(-x-1\right)$$

Factor out common term -x-1 by using distributive property.

$$\left(-x-1\right)\left(x+3\right)$$

Rewrite the complete factored expression.

$$12\left(-x-1\right)\left(x+3\right)$$

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.

$$-12x^{2}-48x-36=0$$

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.

$$x=\frac{-\left(-48\right)±\sqrt{\left(-48\right)^{2}-4\left(-12\right)\left(-36\right)}}{2\left(-12\right)}$$

Square -48.

$$x=\frac{-\left(-48\right)±\sqrt{2304-4\left(-12\right)\left(-36\right)}}{2\left(-12\right)}$$

Multiply -4 times -12.

$$x=\frac{-\left(-48\right)±\sqrt{2304+48\left(-36\right)}}{2\left(-12\right)}$$

Multiply 48 times -36.

$$x=\frac{-\left(-48\right)±\sqrt{2304-1728}}{2\left(-12\right)}$$

Add 2304 to -1728.

$$x=\frac{-\left(-48\right)±\sqrt{576}}{2\left(-12\right)}$$

Take the square root of 576.

$$x=\frac{-\left(-48\right)±24}{2\left(-12\right)}$$

The opposite of -48 is 48.

$$x=\frac{48±24}{2\left(-12\right)}$$

Multiply 2 times -12.

$$x=\frac{48±24}{-24}$$

Now solve the equation x=\frac{48±24}{-24} when ± is plus. Add 48 to 24.

$$x=\frac{72}{-24}$$

Divide 72 by -24.

$$x=-3$$

Now solve the equation x=\frac{48±24}{-24} when ± is minus. Subtract 24 from 48.

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

Divide 24 by -24.

$$x=-1$$

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

$$-12x^{2}-48x-36=-12\left(x-\left(-3\right)\right)\left(x-\left(-1\right)\right)$$

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

$$-12x^{2}-48x-36=-12\left(x+3\right)\left(x+1\right)$$

Evaluate

$$-12\left(x+1\right)\left(x+3\right)$$