Solve -3x^2-21x-18

$$-3 { x }^{ 2 } -21x-18$$

Factor

$$-3\left(x+1\right)\left(x+6\right)$$

Solution Steps

Factor out 3.

$$3\left(-x^{2}-7x-6\right)$$

Consider -x^{2}-7x-6. Factor the expression by grouping. First, the expression needs to be rewritten as -x^{2}+ax+bx-6. To find a and b, set up a system to be solved.

$$a+b=-7 ab=-\left(-6\right)=6$$

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 6.

$$-1,-6 -2,-3$$

Calculate the sum for each pair.

$$-1-6=-7 -2-3=-5$$

The solution is the pair that gives sum -7.

$$a=-1 b=-6$$

Rewrite -x^{2}-7x-6 as \left(-x^{2}-x\right)+\left(-6x-6\right).

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

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

$$x\left(-x-1\right)+6\left(-x-1\right)$$

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

$$\left(-x-1\right)\left(x+6\right)$$

Rewrite the complete factored expression.

$$3\left(-x-1\right)\left(x+6\right)$$

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.

$$-3x^{2}-21x-18=0$$

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.

$$x=\frac{-\left(-21\right)±\sqrt{\left(-21\right)^{2}-4\left(-3\right)\left(-18\right)}}{2\left(-3\right)}$$

Square -21.

$$x=\frac{-\left(-21\right)±\sqrt{441-4\left(-3\right)\left(-18\right)}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-\left(-21\right)±\sqrt{441+12\left(-18\right)}}{2\left(-3\right)}$$

Multiply 12 times -18.

$$x=\frac{-\left(-21\right)±\sqrt{441-216}}{2\left(-3\right)}$$

Add 441 to -216.

$$x=\frac{-\left(-21\right)±\sqrt{225}}{2\left(-3\right)}$$

Take the square root of 225.

$$x=\frac{-\left(-21\right)±15}{2\left(-3\right)}$$

The opposite of -21 is 21.

$$x=\frac{21±15}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{21±15}{-6}$$

Now solve the equation x=\frac{21±15}{-6} when ± is plus. Add 21 to 15.

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

Divide 36 by -6.

$$x=-6$$

Now solve the equation x=\frac{21±15}{-6} when ± is minus. Subtract 15 from 21.

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

Divide 6 by -6.

$$x=-1$$

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 -1 for x_{2}.

$$-3x^{2}-21x-18=-3\left(x-\left(-6\right)\right)\left(x-\left(-1\right)\right)$$

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

$$-3x^{2}-21x-18=-3\left(x+6\right)\left(x+1\right)$$

Evaluate

$$-3\left(x+1\right)\left(x+6\right)$$