Solve x^2-16x+26

$$x ^ { 2 } - 16 x + 26$$

Factor

$$\left(x-\left(8-\sqrt{38}\right)\right)\left(x-\left(\sqrt{38}+8\right)\right)$$

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.

$$x^{2}-16x+26=0$$

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.

$$x=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\times 26}}{2}$$

Square -16.

$$x=\frac{-\left(-16\right)±\sqrt{256-4\times 26}}{2}$$

Multiply -4 times 26.

$$x=\frac{-\left(-16\right)±\sqrt{256-104}}{2}$$

Add 256 to -104.

$$x=\frac{-\left(-16\right)±\sqrt{152}}{2}$$

Take the square root of 152.

$$x=\frac{-\left(-16\right)±2\sqrt{38}}{2}$$

The opposite of -16 is 16.

$$x=\frac{16±2\sqrt{38}}{2}$$

Now solve the equation x=\frac{16±2\sqrt{38}}{2} when ± is plus. Add 16 to 2\sqrt{38}.

$$x=\frac{2\sqrt{38}+16}{2}$$

Divide 16+2\sqrt{38} by 2.

$$x=\sqrt{38}+8$$

Now solve the equation x=\frac{16±2\sqrt{38}}{2} when ± is minus. Subtract 2\sqrt{38} from 16.

$$x=\frac{16-2\sqrt{38}}{2}$$

Divide 16-2\sqrt{38} by 2.

$$x=8-\sqrt{38}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 8+\sqrt{38} for x_{1} and 8-\sqrt{38} for x_{2}.

$$x^{2}-16x+26=\left(x-\left(\sqrt{38}+8\right)\right)\left(x-\left(8-\sqrt{38}\right)\right)$$

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.

$$x ^ 2 -16x +26 = 0$$

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

$$r + s = 16 rs = 26$$

Two numbers r and s sum up to 16 exactly when the average of the two numbers is \frac{1}{2}*16 = 8. 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>

$$r = 8 - u s = 8 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 26

$$(8 - u) (8 + u) = 26$$

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

$$64 - u^2 = 26$$

Simplify the expression by subtracting 64 on both sides

$$-u^2 = 26-64 = -38$$

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

$$u^2 = 38 u = \pm\sqrt{38} = \pm \sqrt{38}$$

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

$$r =8 - \sqrt{38} = 1.836 s = 8 + \sqrt{38} = 14.164$$

Evaluate

$$x^{2}-16x+26$$