Solve b^2-26b+89

$$b ^ { 2 } - 26 b + 89$$

Factor

$$\left(b-\left(13-4\sqrt{5}\right)\right)\left(b-\left(4\sqrt{5}+13\right)\right)$$

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.

$$b^{2}-26b+89=0$$

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.

$$b=\frac{-\left(-26\right)±\sqrt{\left(-26\right)^{2}-4\times 89}}{2}$$

Square -26.

$$b=\frac{-\left(-26\right)±\sqrt{676-4\times 89}}{2}$$

Multiply -4 times 89.

$$b=\frac{-\left(-26\right)±\sqrt{676-356}}{2}$$

Add 676 to -356.

$$b=\frac{-\left(-26\right)±\sqrt{320}}{2}$$

Take the square root of 320.

$$b=\frac{-\left(-26\right)±8\sqrt{5}}{2}$$

The opposite of -26 is 26.

$$b=\frac{26±8\sqrt{5}}{2}$$

Now solve the equation b=\frac{26±8\sqrt{5}}{2} when ± is plus. Add 26 to 8\sqrt{5}.

$$b=\frac{8\sqrt{5}+26}{2}$$

Divide 26+8\sqrt{5} by 2.

$$b=4\sqrt{5}+13$$

Now solve the equation b=\frac{26±8\sqrt{5}}{2} when ± is minus. Subtract 8\sqrt{5} from 26.

$$b=\frac{26-8\sqrt{5}}{2}$$

Divide 26-8\sqrt{5} by 2.

$$b=13-4\sqrt{5}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 13+4\sqrt{5} for x_{1} and 13-4\sqrt{5} for x_{2}.

$$b^{2}-26b+89=\left(b-\left(4\sqrt{5}+13\right)\right)\left(b-\left(13-4\sqrt{5}\right)\right)$$

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.

$$x ^ 2 -26x +89 = 0$$

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

$$r + s = 26 rs = 89$$

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 = 89

$$(13 - u) (13 + u) = 89$$

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

$$169 - u^2 = 89$$

Simplify the expression by subtracting 169 on both sides

$$-u^2 = 89-169 = -80$$

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

$$u^2 = 80 u = \pm\sqrt{80} = \pm \sqrt{80}$$

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

$$r =13 - \sqrt{80} = 4.056 s = 13 + \sqrt{80} = 21.944$$

Evaluate

$$b^{2}-26b+89$$