Solve x^2-298x+500

$$x ^ { 2 } - 298 x + 500$$

Factor

$$\left(x-\left(149-\sqrt{21701}\right)\right)\left(x-\left(\sqrt{21701}+149\right)\right)$$

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.

$$x^{2}-298x+500=0$$

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.

$$x=\frac{-\left(-298\right)±\sqrt{\left(-298\right)^{2}-4\times 500}}{2}$$

Square -298.

$$x=\frac{-\left(-298\right)±\sqrt{88804-4\times 500}}{2}$$

Multiply -4 times 500.

$$x=\frac{-\left(-298\right)±\sqrt{88804-2000}}{2}$$

Add 88804 to -2000.

$$x=\frac{-\left(-298\right)±\sqrt{86804}}{2}$$

Take the square root of 86804.

$$x=\frac{-\left(-298\right)±2\sqrt{21701}}{2}$$

The opposite of -298 is 298.

$$x=\frac{298±2\sqrt{21701}}{2}$$

Now solve the equation x=\frac{298±2\sqrt{21701}}{2} when ± is plus. Add 298 to 2\sqrt{21701}.

$$x=\frac{2\sqrt{21701}+298}{2}$$

Divide 298+2\sqrt{21701} by 2.

$$x=\sqrt{21701}+149$$

Now solve the equation x=\frac{298±2\sqrt{21701}}{2} when ± is minus. Subtract 2\sqrt{21701} from 298.

$$x=\frac{298-2\sqrt{21701}}{2}$$

Divide 298-2\sqrt{21701} by 2.

$$x=149-\sqrt{21701}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 149+\sqrt{21701} for x_{1} and 149-\sqrt{21701} for x_{2}.

$$x^{2}-298x+500=\left(x-\left(\sqrt{21701}+149\right)\right)\left(x-\left(149-\sqrt{21701}\right)\right)$$

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.

$$x ^ 2 -298x +500 = 0$$

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

$$r + s = 298 rs = 500$$

Two numbers r and s sum up to 298 exactly when the average of the two numbers is \frac{1}{2}*298 = 149. 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>

$$r = 149 - u s = 149 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 500

$$(149 - u) (149 + u) = 500$$

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

$$22201 - u^2 = 500$$

Simplify the expression by subtracting 22201 on both sides

$$-u^2 = 500-22201 = -21701$$

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

$$u^2 = 21701 u = \pm\sqrt{21701} = \pm \sqrt{21701}$$

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

$$r =149 - \sqrt{21701} = 1.687 s = 149 + \sqrt{21701} = 296.313$$

Evaluate

$$x^{2}-298x+500$$