Solve x^2-130x+1800=0

$$x ^ { 2 } - 130 x + 1800 = 0$$

Solve for x

$$x=5\sqrt{97}+65\approx 114.244289009$$

$$x=65-5\sqrt{97}\approx 15.755710991$$

Steps Using the Quadratic Formula

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.

$$x^{2}-130x+1800=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -130 for b, and 1800 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-130\right)±\sqrt{\left(-130\right)^{2}-4\times 1800}}{2}$$

Square -130.

$$x=\frac{-\left(-130\right)±\sqrt{16900-4\times 1800}}{2}$$

Multiply -4 times 1800.

$$x=\frac{-\left(-130\right)±\sqrt{16900-7200}}{2}$$

Add 16900 to -7200.

$$x=\frac{-\left(-130\right)±\sqrt{9700}}{2}$$

Take the square root of 9700.

$$x=\frac{-\left(-130\right)±10\sqrt{97}}{2}$$

The opposite of -130 is 130.

$$x=\frac{130±10\sqrt{97}}{2}$$

Now solve the equation x=\frac{130±10\sqrt{97}}{2} when ± is plus. Add 130 to 10\sqrt{97}.

$$x=\frac{10\sqrt{97}+130}{2}$$

Divide 130+10\sqrt{97} by 2.

$$x=5\sqrt{97}+65$$

Now solve the equation x=\frac{130±10\sqrt{97}}{2} when ± is minus. Subtract 10\sqrt{97} from 130.

$$x=\frac{130-10\sqrt{97}}{2}$$

Divide 130-10\sqrt{97} by 2.

$$x=65-5\sqrt{97}$$

The equation is now solved.

$$x=5\sqrt{97}+65 x=65-5\sqrt{97}$$

Steps for Completing the Square

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$x^{2}-130x+1800=0$$

Subtract 1800 from both sides of the equation.

$$x^{2}-130x+1800-1800=-1800$$

Subtracting 1800 from itself leaves 0.

$$x^{2}-130x=-1800$$

Divide -130, the coefficient of the x term, by 2 to get -65. Then add the square of -65 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-130x+\left(-65\right)^{2}=-1800+\left(-65\right)^{2}$$

Square -65.

$$x^{2}-130x+4225=-1800+4225$$

Add -1800 to 4225.

$$x^{2}-130x+4225=2425$$

Factor x^{2}-130x+4225. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-65\right)^{2}=2425$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-65\right)^{2}}=\sqrt{2425}$$

Simplify.

$$x-65=5\sqrt{97} x-65=-5\sqrt{97}$$

Add 65 to both sides of the equation.

$$x=5\sqrt{97}+65 x=65-5\sqrt{97}$$

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.

$$x ^ 2 -130x +1800 = 0$$

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

$$r + s = 130 rs = 1800$$

Two numbers r and s sum up to 130 exactly when the average of the two numbers is \frac{1}{2}*130 = 65. 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>

$$r = 65 - u s = 65 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 1800

$$(65 - u) (65 + u) = 1800$$

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

$$4225 - u^2 = 1800$$

Simplify the expression by subtracting 4225 on both sides

$$-u^2 = 1800-4225 = -2425$$

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

$$u^2 = 2425 u = \pm\sqrt{2425} = \pm \sqrt{2425}$$

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

$$r =65 - \sqrt{2425} = 15.756 s = 65 + \sqrt{2425} = 114.244$$