Solve -x^2-10x-21

$$- { x }^{ 2 } -10x-21$$

Factor

$$-\left(x+3\right)\left(x+7\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as -x^{2}+ax+bx-21. To find a and b, set up a system to be solved.

$$a+b=-10 ab=-\left(-21\right)=21$$

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 21.

$$-1,-21 -3,-7$$

Calculate the sum for each pair.

$$-1-21=-22 -3-7=-10$$

The solution is the pair that gives sum -10.

$$a=-3 b=-7$$

Rewrite -x^{2}-10x-21 as \left(-x^{2}-3x\right)+\left(-7x-21\right).

$$\left(-x^{2}-3x\right)+\left(-7x-21\right)$$

Factor out x in the first and 7 in the second group.

$$x\left(-x-3\right)+7\left(-x-3\right)$$

Factor out common term -x-3 by using distributive property.

$$\left(-x-3\right)\left(x+7\right)$$

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.

$$-x^{2}-10x-21=0$$

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.

$$x=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\left(-1\right)\left(-21\right)}}{2\left(-1\right)}$$

Square -10.

$$x=\frac{-\left(-10\right)±\sqrt{100-4\left(-1\right)\left(-21\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-10\right)±\sqrt{100+4\left(-21\right)}}{2\left(-1\right)}$$

Multiply 4 times -21.

$$x=\frac{-\left(-10\right)±\sqrt{100-84}}{2\left(-1\right)}$$

Add 100 to -84.

$$x=\frac{-\left(-10\right)±\sqrt{16}}{2\left(-1\right)}$$

Take the square root of 16.

$$x=\frac{-\left(-10\right)±4}{2\left(-1\right)}$$

The opposite of -10 is 10.

$$x=\frac{10±4}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{10±4}{-2}$$

Now solve the equation x=\frac{10±4}{-2} when ± is plus. Add 10 to 4.

$$x=\frac{14}{-2}$$

Divide 14 by -2.

$$x=-7$$

Now solve the equation x=\frac{10±4}{-2} when ± is minus. Subtract 4 from 10.

$$x=\frac{6}{-2}$$

Divide 6 by -2.

$$x=-3$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -7 for x_{1} and -3 for x_{2}.

$$-x^{2}-10x-21=-\left(x-\left(-7\right)\right)\left(x-\left(-3\right)\right)$$

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

$$-x^{2}-10x-21=-\left(x+7\right)\left(x+3\right)$$

Evaluate

$$-\left(x+3\right)\left(x+7\right)$$