Solve x^2-4x-21

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

Factor

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

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.

$$a+b=-4 ab=1\left(-21\right)=-21$$

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.

$$a=-7 b=3$$

Rewrite x^{2}-4x-21 as \left(x^{2}-7x\right)+\left(3x-21\right).

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

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

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

Factor out common term x-7 by using distributive property.

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

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.

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

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.

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

Square -4.

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

Multiply -4 times -21.

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

Add 16 to 84.

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

Take the square root of 100.

$$x=\frac{-\left(-4\right)±10}{2}$$

The opposite of -4 is 4.

$$x=\frac{4±10}{2}$$

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}.

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

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

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