Solve x^2-46x+460

$$x ^ { 2 } - 46 x + 460$$

Factor

$$\left(x-\left(23-\sqrt{69}\right)\right)\left(x-\left(\sqrt{69}+23\right)\right)$$

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.

$$x^{2}-46x+460=0$$

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.

$$x=\frac{-\left(-46\right)±\sqrt{\left(-46\right)^{2}-4\times 460}}{2}$$

Square -46.

$$x=\frac{-\left(-46\right)±\sqrt{2116-4\times 460}}{2}$$

Multiply -4 times 460.

$$x=\frac{-\left(-46\right)±\sqrt{2116-1840}}{2}$$

Add 2116 to -1840.

$$x=\frac{-\left(-46\right)±\sqrt{276}}{2}$$

Take the square root of 276.

$$x=\frac{-\left(-46\right)±2\sqrt{69}}{2}$$

The opposite of -46 is 46.

$$x=\frac{46±2\sqrt{69}}{2}$$

Now solve the equation x=\frac{46±2\sqrt{69}}{2} when ± is plus. Add 46 to 2\sqrt{69}.

$$x=\frac{2\sqrt{69}+46}{2}$$

Divide 46+2\sqrt{69} by 2.

$$x=\sqrt{69}+23$$

Now solve the equation x=\frac{46±2\sqrt{69}}{2} when ± is minus. Subtract 2\sqrt{69} from 46.

$$x=\frac{46-2\sqrt{69}}{2}$$

Divide 46-2\sqrt{69} by 2.

$$x=23-\sqrt{69}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 23+\sqrt{69} for x_{1} and 23-\sqrt{69} for x_{2}.

$$x^{2}-46x+460=\left(x-\left(\sqrt{69}+23\right)\right)\left(x-\left(23-\sqrt{69}\right)\right)$$

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.

$$x ^ 2 -46x +460 = 0$$

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

$$r + s = 46 rs = 460$$

Two numbers r and s sum up to 46 exactly when the average of the two numbers is \frac{1}{2}*46 = 23. 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>

$$r = 23 - u s = 23 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 460

$$(23 - u) (23 + u) = 460$$

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

$$529 - u^2 = 460$$

Simplify the expression by subtracting 529 on both sides

$$-u^2 = 460-529 = -69$$

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

$$u^2 = 69 u = \pm\sqrt{69} = \pm \sqrt{69}$$

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

$$r =23 - \sqrt{69} = 14.693 s = 23 + \sqrt{69} = 31.307$$

Evaluate

$$x^{2}-46x+460$$