Solve -x^2+26x-1

$$- x ^ { 2 } + 26 x - 1$$

Factor

$$-\left(x-\left(13-2\sqrt{42}\right)\right)\left(x-\left(2\sqrt{42}+13\right)\right)$$

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.

$$-x^{2}+26x-1=0$$

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.

$$x=\frac{-26±\sqrt{26^{2}-4\left(-1\right)\left(-1\right)}}{2\left(-1\right)}$$

Square 26.

$$x=\frac{-26±\sqrt{676-4\left(-1\right)\left(-1\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-26±\sqrt{676+4\left(-1\right)}}{2\left(-1\right)}$$

Multiply 4 times -1.

$$x=\frac{-26±\sqrt{676-4}}{2\left(-1\right)}$$

Add 676 to -4.

$$x=\frac{-26±\sqrt{672}}{2\left(-1\right)}$$

Take the square root of 672.

$$x=\frac{-26±4\sqrt{42}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-26±4\sqrt{42}}{-2}$$

Now solve the equation x=\frac{-26±4\sqrt{42}}{-2} when ± is plus. Add -26 to 4\sqrt{42}.

$$x=\frac{4\sqrt{42}-26}{-2}$$

Divide -26+4\sqrt{42} by -2.

$$x=13-2\sqrt{42}$$

Now solve the equation x=\frac{-26±4\sqrt{42}}{-2} when ± is minus. Subtract 4\sqrt{42} from -26.

$$x=\frac{-4\sqrt{42}-26}{-2}$$

Divide -26-4\sqrt{42} by -2.

$$x=2\sqrt{42}+13$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 13-2\sqrt{42} for x_{1} and 13+2\sqrt{42} for x_{2}.

$$-x^{2}+26x-1=-\left(x-\left(13-2\sqrt{42}\right)\right)\left(x-\left(2\sqrt{42}+13\right)\right)$$

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.

$$x ^ 2 -26x +1 = 0$$

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

$$r + s = 26 rs = 1$$

Two numbers r and s sum up to 26 exactly when the average of the two numbers is \frac{1}{2}*26 = 13. 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>

$$r = 13 - u s = 13 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 1

$$(13 - u) (13 + u) = 1$$

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

$$169 - u^2 = 1$$

Simplify the expression by subtracting 169 on both sides

$$-u^2 = 1-169 = -168$$

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

$$u^2 = 168 u = \pm\sqrt{168} = \pm \sqrt{168}$$

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

$$r =13 - \sqrt{168} = 0.039 s = 13 + \sqrt{168} = 25.961$$

Evaluate

$$-x^{2}+26x-1$$