Solve x^2-58x+121=0

$$x ^ { 2 } - 58 x + 121 = 0$$

Solve for x

$$x=12\sqrt{5}+29\approx 55.83281573$$

$$x=29-12\sqrt{5}\approx 2.16718427$$

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.

$$x^{2}-58x+121=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -58 for b, and 121 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-58\right)±\sqrt{\left(-58\right)^{2}-4\times 121}}{2}$$

Square -58.

$$x=\frac{-\left(-58\right)±\sqrt{3364-4\times 121}}{2}$$

Multiply -4 times 121.

$$x=\frac{-\left(-58\right)±\sqrt{3364-484}}{2}$$

Add 3364 to -484.

$$x=\frac{-\left(-58\right)±\sqrt{2880}}{2}$$

Take the square root of 2880.

$$x=\frac{-\left(-58\right)±24\sqrt{5}}{2}$$

The opposite of -58 is 58.

$$x=\frac{58±24\sqrt{5}}{2}$$

Now solve the equation x=\frac{58±24\sqrt{5}}{2} when ± is plus. Add 58 to 24\sqrt{5}.

$$x=\frac{24\sqrt{5}+58}{2}$$

Divide 58+24\sqrt{5} by 2.

$$x=12\sqrt{5}+29$$

Now solve the equation x=\frac{58±24\sqrt{5}}{2} when ± is minus. Subtract 24\sqrt{5} from 58.

$$x=\frac{58-24\sqrt{5}}{2}$$

Divide 58-24\sqrt{5} by 2.

$$x=29-12\sqrt{5}$$

The equation is now solved.

$$x=12\sqrt{5}+29 x=29-12\sqrt{5}$$

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.

$$x^{2}-58x+121=0$$

Subtract 121 from both sides of the equation.

$$x^{2}-58x+121-121=-121$$

Subtracting 121 from itself leaves 0.

$$x^{2}-58x=-121$$

Divide -58, the coefficient of the x term, by 2 to get -29. Then add the square of -29 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-58x+\left(-29\right)^{2}=-121+\left(-29\right)^{2}$$

Square -29.

$$x^{2}-58x+841=-121+841$$

Add -121 to 841.

$$x^{2}-58x+841=720$$

Factor x^{2}-58x+841. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-29\right)^{2}=720$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-29\right)^{2}}=\sqrt{720}$$

Simplify.

$$x-29=12\sqrt{5} x-29=-12\sqrt{5}$$

Add 29 to both sides of the equation.

$$x=12\sqrt{5}+29 x=29-12\sqrt{5}$$

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.

$$x ^ 2 -58x +121 = 0$$

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

$$r + s = 58 rs = 121$$

Two numbers r and s sum up to 58 exactly when the average of the two numbers is \frac{1}{2}*58 = 29. 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>

$$r = 29 - u s = 29 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 121

$$(29 - u) (29 + u) = 121$$

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

$$841 - u^2 = 121$$

Simplify the expression by subtracting 841 on both sides

$$-u^2 = 121-841 = -720$$

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

$$u^2 = 720 u = \pm\sqrt{720} = \pm \sqrt{720}$$

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

$$r =29 - \sqrt{720} = 2.167 s = 29 + \sqrt{720} = 55.833$$