Solve x^2+2x-399

$$x ^ { 2 } + 2 x - 399$$

Factor

$$\left(x-19\right)\left(x+21\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-399. To find a and b, set up a system to be solved.

$$a+b=2 ab=1\left(-399\right)=-399$$

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 -399.

$$-1,399 -3,133 -7,57 -19,21$$

Calculate the sum for each pair.

$$-1+399=398 -3+133=130 -7+57=50 -19+21=2$$

The solution is the pair that gives sum 2.

$$a=-19 b=21$$

Rewrite x^{2}+2x-399 as \left(x^{2}-19x\right)+\left(21x-399\right).

$$\left(x^{2}-19x\right)+\left(21x-399\right)$$

Factor out x in the first and 21 in the second group.

$$x\left(x-19\right)+21\left(x-19\right)$$

Factor out common term x-19 by using distributive property.

$$\left(x-19\right)\left(x+21\right)$$

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.

$$x^{2}+2x-399=0$$

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.

$$x=\frac{-2±\sqrt{2^{2}-4\left(-399\right)}}{2}$$

Square 2.

$$x=\frac{-2±\sqrt{4-4\left(-399\right)}}{2}$$

Multiply -4 times -399.

$$x=\frac{-2±\sqrt{4+1596}}{2}$$

Add 4 to 1596.

$$x=\frac{-2±\sqrt{1600}}{2}$$

Take the square root of 1600.

$$x=\frac{-2±40}{2}$$

Now solve the equation x=\frac{-2±40}{2} when ± is plus. Add -2 to 40.

$$x=\frac{38}{2}$$

Divide 38 by 2.

$$x=19$$

Now solve the equation x=\frac{-2±40}{2} when ± is minus. Subtract 40 from -2.

$$x=-\frac{42}{2}$$

Divide -42 by 2.

$$x=-21$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 19 for x_{1} and -21 for x_{2}.

$$x^{2}+2x-399=\left(x-19\right)\left(x-\left(-21\right)\right)$$

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

$$x^{2}+2x-399=\left(x-19\right)\left(x+21\right)$$

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.

$$x ^ 2 +2x -399 = 0$$

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

$$r + s = -2 rs = -399$$

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

$$r = -1 - u s = -1 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -399

$$(-1 - u) (-1 + u) = -399$$

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

$$1 - u^2 = -399$$

Simplify the expression by subtracting 1 on both sides

$$-u^2 = -399-1 = -400$$

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

$$u^2 = 400 u = \pm\sqrt{400} = \pm 20$$

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

$$r =-1 - 20 = -21 s = -1 + 20 = 19$$

Evaluate

$$\left(x-19\right)\left(x+21\right)$$