Solve x^2+4x-21

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

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.

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

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. 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.

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

Square 4.

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

Multiply -4 times -21.

$$x=\frac{-4±\sqrt{16+84}}{2}$$

Add 16 to 84.

$$x=\frac{-4±\sqrt{100}}{2}$$

Take the square root of 100.

$$x=\frac{-4±10}{2}$$

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

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

Divide 6 by 2.

$$x=3$$

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

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

Divide -14 by 2.

$$x=-7$$

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

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

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

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

Steps Using Direct Factoring Method

Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form x^2+Bx+C=0.

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

Let r and s be the factors for the quadratic equation such that x^2+Bx+C=(x−r)(x−s) where sum of factors (r+s)=−B and the product of factors rs = C

$$r + s = -4 rs = -21$$

Two numbers r and s sum up to -4 exactly when the average of the two numbers is \frac{1}{2}*-4 = -2. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented by the quadratic equation y=x^2+Bx+C. The values of r and s are equidistant from the center by an unknown quantity u. Express r and s with respect to variable u. <div style='padding: 8px'><img src='https://opalmath.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = -2 - u s = -2 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -21

$$(-2 - u) (-2 + u) = -21$$

Simplify by expanding (a -b) (a + b) = a^2 – b^2

$$4 - u^2 = -21$$

Simplify the expression by subtracting 4 on both sides

$$-u^2 = -21-4 = -25$$

Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u

$$u^2 = 25 u = \pm\sqrt{25} = \pm 5$$

The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.

$$r =-2 - 5 = -7 s = -2 + 5 = 3$$

Evaluate

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