Solve x^2-4x-221

$$x ^ { 2 } - 4 x - 221$$

Factor

$$\left(x-17\right)\left(x+13\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-221. To find a and b, set up a system to be solved.

$$a+b=-4 ab=1\left(-221\right)=-221$$

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 -221.

$$1,-221 13,-17$$

Calculate the sum for each pair.

$$1-221=-220 13-17=-4$$

The solution is the pair that gives sum -4.

$$a=-17 b=13$$

Rewrite x^{2}-4x-221 as \left(x^{2}-17x\right)+\left(13x-221\right).

$$\left(x^{2}-17x\right)+\left(13x-221\right)$$

Factor out x in the first and 13 in the second group.

$$x\left(x-17\right)+13\left(x-17\right)$$

Factor out common term x-17 by using distributive property.

$$\left(x-17\right)\left(x+13\right)$$

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.

$$x^{2}-4x-221=0$$

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.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-221\right)}}{2}$$

Square -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\left(-221\right)}}{2}$$

Multiply -4 times -221.

$$x=\frac{-\left(-4\right)±\sqrt{16+884}}{2}$$

Add 16 to 884.

$$x=\frac{-\left(-4\right)±\sqrt{900}}{2}$$

Take the square root of 900.

$$x=\frac{-\left(-4\right)±30}{2}$$

The opposite of -4 is 4.

$$x=\frac{4±30}{2}$$

Now solve the equation x=\frac{4±30}{2} when ± is plus. Add 4 to 30.

$$x=\frac{34}{2}$$

Divide 34 by 2.

$$x=17$$

Now solve the equation x=\frac{4±30}{2} when ± is minus. Subtract 30 from 4.

$$x=-\frac{26}{2}$$

Divide -26 by 2.

$$x=-13$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 17 for x_{1} and -13 for x_{2}.

$$x^{2}-4x-221=\left(x-17\right)\left(x-\left(-13\right)\right)$$

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

$$x^{2}-4x-221=\left(x-17\right)\left(x+13\right)$$

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.

$$x ^ 2 -4x -221 = 0$$

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

$$r + s = 4 rs = -221$$

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 = -221

$$(2 - u) (2 + u) = -221$$

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

$$4 - u^2 = -221$$

Simplify the expression by subtracting 4 on both sides

$$-u^2 = -221-4 = -225$$

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

$$u^2 = 225 u = \pm\sqrt{225} = \pm 15$$

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

$$r =2 - 15 = -13 s = 2 + 15 = 17$$

Evaluate

$$\left(x-17\right)\left(x+13\right)$$