Solve 21-4x-x^2

$$21 - 4 x - x ^ { 2 }$$

Factor

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

Solution Steps

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-x^{2}-4x+21$$

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.

$$a+b=-4 ab=-21=-21$$

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 -21.

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

Calculate the sum for each pair.

$$1-21=-20 3-7=-4$$

The solution is the pair that gives sum -4.

$$a=3 b=-7$$

Rewrite -x^{2}-4x+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.

$$-x^{2}-4x+21=0$$

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.

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

Square -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\left(-1\right)\times 21}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-4\right)±\sqrt{16+4\times 21}}{2\left(-1\right)}$$

Multiply 4 times 21.

$$x=\frac{-\left(-4\right)±\sqrt{16+84}}{2\left(-1\right)}$$

Add 16 to 84.

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

Take the square root of 100.

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

The opposite of -4 is 4.

$$x=\frac{4±10}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{4±10}{-2}$$

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

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

Divide 14 by -2.

$$x=-7$$

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

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

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}.

$$-x^{2}-4x+21=-\left(x-\left(-7\right)\right)\left(x-3\right)$$

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

$$-x^{2}-4x+21=-\left(x+7\right)\left(x-3\right)$$

Evaluate

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