Solve x^2+2x-98

$$x ^ { 2 } + 2 x - 98$$

Factor

$$\left(x-\left(-3\sqrt{11}-1\right)\right)\left(x-\left(3\sqrt{11}-1\right)\right)$$

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.

$$x^{2}+2x-98=0$$

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.

$$x=\frac{-2±\sqrt{2^{2}-4\left(-98\right)}}{2}$$

Square 2.

$$x=\frac{-2±\sqrt{4-4\left(-98\right)}}{2}$$

Multiply -4 times -98.

$$x=\frac{-2±\sqrt{4+392}}{2}$$

Add 4 to 392.

$$x=\frac{-2±\sqrt{396}}{2}$$

Take the square root of 396.

$$x=\frac{-2±6\sqrt{11}}{2}$$

Now solve the equation x=\frac{-2±6\sqrt{11}}{2} when ± is plus. Add -2 to 6\sqrt{11}.

$$x=\frac{6\sqrt{11}-2}{2}$$

Divide -2+6\sqrt{11} by 2.

$$x=3\sqrt{11}-1$$

Now solve the equation x=\frac{-2±6\sqrt{11}}{2} when ± is minus. Subtract 6\sqrt{11} from -2.

$$x=\frac{-6\sqrt{11}-2}{2}$$

Divide -2-6\sqrt{11} by 2.

$$x=-3\sqrt{11}-1$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -1+3\sqrt{11} for x_{1} and -1-3\sqrt{11} for x_{2}.

$$x^{2}+2x-98=\left(x-\left(3\sqrt{11}-1\right)\right)\left(x-\left(-3\sqrt{11}-1\right)\right)$$

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.

$$x ^ 2 +2x -98 = 0$$

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

$$r + s = -2 rs = -98$$

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 = -98

$$(-1 - u) (-1 + u) = -98$$

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

$$1 - u^2 = -98$$

Simplify the expression by subtracting 1 on both sides

$$-u^2 = -98-1 = -99$$

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

$$u^2 = 99 u = \pm\sqrt{99} = \pm \sqrt{99}$$

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

$$r =-1 - \sqrt{99} = -10.950 s = -1 + \sqrt{99} = 8.950$$

Evaluate

$$x^{2}+2x-98$$