Solve -3x^2-30x-72

$$-3 { x }^{ 2 } -30x-72$$

Factor

$$-3\left(x+4\right)\left(x+6\right)$$

Solution Steps

Factor out 3.

$$3\left(-x^{2}-10x-24\right)$$

Consider -x^{2}-10x-24. Factor the expression by grouping. First, the expression needs to be rewritten as -x^{2}+ax+bx-24. To find a and b, set up a system to be solved.

$$a+b=-10 ab=-\left(-24\right)=24$$

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 24.

$$-1,-24 -2,-12 -3,-8 -4,-6$$

Calculate the sum for each pair.

$$-1-24=-25 -2-12=-14 -3-8=-11 -4-6=-10$$

The solution is the pair that gives sum -10.

$$a=-4 b=-6$$

Rewrite -x^{2}-10x-24 as \left(-x^{2}-4x\right)+\left(-6x-24\right).

$$\left(-x^{2}-4x\right)+\left(-6x-24\right)$$

Factor out x in the first and 6 in the second group.

$$x\left(-x-4\right)+6\left(-x-4\right)$$

Factor out common term -x-4 by using distributive property.

$$\left(-x-4\right)\left(x+6\right)$$

Rewrite the complete factored expression.

$$3\left(-x-4\right)\left(x+6\right)$$

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.

$$-3x^{2}-30x-72=0$$

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.

$$x=\frac{-\left(-30\right)±\sqrt{\left(-30\right)^{2}-4\left(-3\right)\left(-72\right)}}{2\left(-3\right)}$$

Square -30.

$$x=\frac{-\left(-30\right)±\sqrt{900-4\left(-3\right)\left(-72\right)}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-\left(-30\right)±\sqrt{900+12\left(-72\right)}}{2\left(-3\right)}$$

Multiply 12 times -72.

$$x=\frac{-\left(-30\right)±\sqrt{900-864}}{2\left(-3\right)}$$

Add 900 to -864.

$$x=\frac{-\left(-30\right)±\sqrt{36}}{2\left(-3\right)}$$

Take the square root of 36.

$$x=\frac{-\left(-30\right)±6}{2\left(-3\right)}$$

The opposite of -30 is 30.

$$x=\frac{30±6}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{30±6}{-6}$$

Now solve the equation x=\frac{30±6}{-6} when ± is plus. Add 30 to 6.

$$x=\frac{36}{-6}$$

Divide 36 by -6.

$$x=-6$$

Now solve the equation x=\frac{30±6}{-6} when ± is minus. Subtract 6 from 30.

$$x=\frac{24}{-6}$$

Divide 24 by -6.

$$x=-4$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -6 for x_{1} and -4 for x_{2}.

$$-3x^{2}-30x-72=-3\left(x-\left(-6\right)\right)\left(x-\left(-4\right)\right)$$

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

$$-3x^{2}-30x-72=-3\left(x+6\right)\left(x+4\right)$$

Evaluate

$$-3\left(x+4\right)\left(x+6\right)$$