Solve x^2-226x+1103=0

$$x ^ { 2 } - 226 x + 1103 = 0$$

Solve for x

$$x=\sqrt{11666}+113\approx 221.009258862$$

$$x=113-\sqrt{11666}\approx 4.990741138$$

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.

$$x^{2}-226x+1103=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -226 for b, and 1103 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-226\right)±\sqrt{\left(-226\right)^{2}-4\times 1103}}{2}$$

Square -226.

$$x=\frac{-\left(-226\right)±\sqrt{51076-4\times 1103}}{2}$$

Multiply -4 times 1103.

$$x=\frac{-\left(-226\right)±\sqrt{51076-4412}}{2}$$

Add 51076 to -4412.

$$x=\frac{-\left(-226\right)±\sqrt{46664}}{2}$$

Take the square root of 46664.

$$x=\frac{-\left(-226\right)±2\sqrt{11666}}{2}$$

The opposite of -226 is 226.

$$x=\frac{226±2\sqrt{11666}}{2}$$

Now solve the equation x=\frac{226±2\sqrt{11666}}{2} when ± is plus. Add 226 to 2\sqrt{11666}.

$$x=\frac{2\sqrt{11666}+226}{2}$$

Divide 226+2\sqrt{11666} by 2.

$$x=\sqrt{11666}+113$$

Now solve the equation x=\frac{226±2\sqrt{11666}}{2} when ± is minus. Subtract 2\sqrt{11666} from 226.

$$x=\frac{226-2\sqrt{11666}}{2}$$

Divide 226-2\sqrt{11666} by 2.

$$x=113-\sqrt{11666}$$

The equation is now solved.

$$x=\sqrt{11666}+113 x=113-\sqrt{11666}$$

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.

$$x^{2}-226x+1103=0$$

Subtract 1103 from both sides of the equation.

$$x^{2}-226x+1103-1103=-1103$$

Subtracting 1103 from itself leaves 0.

$$x^{2}-226x=-1103$$

Divide -226, the coefficient of the x term, by 2 to get -113. Then add the square of -113 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-226x+\left(-113\right)^{2}=-1103+\left(-113\right)^{2}$$

Square -113.

$$x^{2}-226x+12769=-1103+12769$$

Add -1103 to 12769.

$$x^{2}-226x+12769=11666$$

Factor x^{2}-226x+12769. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-113\right)^{2}=11666$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-113\right)^{2}}=\sqrt{11666}$$

Simplify.

$$x-113=\sqrt{11666} x-113=-\sqrt{11666}$$

Add 113 to both sides of the equation.

$$x=\sqrt{11666}+113 x=113-\sqrt{11666}$$

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.

$$x ^ 2 -226x +1103 = 0$$

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

$$r + s = 226 rs = 1103$$

Two numbers r and s sum up to 226 exactly when the average of the two numbers is \frac{1}{2}*226 = 113. 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>

$$r = 113 - u s = 113 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 1103

$$(113 - u) (113 + u) = 1103$$

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

$$12769 - u^2 = 1103$$

Simplify the expression by subtracting 12769 on both sides

$$-u^2 = 1103-12769 = -11666$$

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

$$u^2 = 11666 u = \pm\sqrt{11666} = \pm \sqrt{11666}$$

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

$$r =113 - \sqrt{11666} = 4.991 s = 113 + \sqrt{11666} = 221.009$$