Solve x^2+20x-224

$$x ^ { 2 } + 20 x - 224$$

Factor

$$\left(x-8\right)\left(x+28\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-224. To find a and b, set up a system to be solved.

$$a+b=20 ab=1\left(-224\right)=-224$$

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 -224.

$$-1,224 -2,112 -4,56 -7,32 -8,28 -14,16$$

Calculate the sum for each pair.

$$-1+224=223 -2+112=110 -4+56=52 -7+32=25 -8+28=20 -14+16=2$$

The solution is the pair that gives sum 20.

$$a=-8 b=28$$

Rewrite x^{2}+20x-224 as \left(x^{2}-8x\right)+\left(28x-224\right).

$$\left(x^{2}-8x\right)+\left(28x-224\right)$$

Factor out x in the first and 28 in the second group.

$$x\left(x-8\right)+28\left(x-8\right)$$

Factor out common term x-8 by using distributive property.

$$\left(x-8\right)\left(x+28\right)$$

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.

$$x^{2}+20x-224=0$$

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.

$$x=\frac{-20±\sqrt{20^{2}-4\left(-224\right)}}{2}$$

Square 20.

$$x=\frac{-20±\sqrt{400-4\left(-224\right)}}{2}$$

Multiply -4 times -224.

$$x=\frac{-20±\sqrt{400+896}}{2}$$

Add 400 to 896.

$$x=\frac{-20±\sqrt{1296}}{2}$$

Take the square root of 1296.

$$x=\frac{-20±36}{2}$$

Now solve the equation x=\frac{-20±36}{2} when ± is plus. Add -20 to 36.

$$x=\frac{16}{2}$$

Divide 16 by 2.

$$x=8$$

Now solve the equation x=\frac{-20±36}{2} when ± is minus. Subtract 36 from -20.

$$x=-\frac{56}{2}$$

Divide -56 by 2.

$$x=-28$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 8 for x_{1} and -28 for x_{2}.

$$x^{2}+20x-224=\left(x-8\right)\left(x-\left(-28\right)\right)$$

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

$$x^{2}+20x-224=\left(x-8\right)\left(x+28\right)$$

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.

$$x ^ 2 +20x -224 = 0$$

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

$$r + s = -20 rs = -224$$

Two numbers r and s sum up to -20 exactly when the average of the two numbers is \frac{1}{2}*-20 = -10. 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>

$$r = -10 - u s = -10 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -224

$$(-10 - u) (-10 + u) = -224$$

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

$$100 - u^2 = -224$$

Simplify the expression by subtracting 100 on both sides

$$-u^2 = -224-100 = -324$$

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

$$u^2 = 324 u = \pm\sqrt{324} = \pm 18$$

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

$$r =-10 - 18 = -28 s = -10 + 18 = 8$$

Evaluate

$$\left(x-8\right)\left(x+28\right)$$