Solve d^2-75d+108

$$d ^ { 2 } - 75 d + 108$$

Factor

$$\left(d-\frac{75-3\sqrt{577}}{2}\right)\left(d-\frac{3\sqrt{577}+75}{2}\right)$$

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.

$$d^{2}-75d+108=0$$

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.

$$d=\frac{-\left(-75\right)±\sqrt{\left(-75\right)^{2}-4\times 108}}{2}$$

Square -75.

$$d=\frac{-\left(-75\right)±\sqrt{5625-4\times 108}}{2}$$

Multiply -4 times 108.

$$d=\frac{-\left(-75\right)±\sqrt{5625-432}}{2}$$

Add 5625 to -432.

$$d=\frac{-\left(-75\right)±\sqrt{5193}}{2}$$

Take the square root of 5193.

$$d=\frac{-\left(-75\right)±3\sqrt{577}}{2}$$

The opposite of -75 is 75.

$$d=\frac{75±3\sqrt{577}}{2}$$

Now solve the equation d=\frac{75±3\sqrt{577}}{2} when ± is plus. Add 75 to 3\sqrt{577}.

$$d=\frac{3\sqrt{577}+75}{2}$$

Now solve the equation d=\frac{75±3\sqrt{577}}{2} when ± is minus. Subtract 3\sqrt{577} from 75.

$$d=\frac{75-3\sqrt{577}}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{75+3\sqrt{577}}{2} for x_{1} and \frac{75-3\sqrt{577}}{2} for x_{2}.

$$d^{2}-75d+108=\left(d-\frac{3\sqrt{577}+75}{2}\right)\left(d-\frac{75-3\sqrt{577}}{2}\right)$$

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.

$$x ^ 2 -75x +108 = 0$$

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

$$r + s = 75 rs = 108$$

Two numbers r and s sum up to 75 exactly when the average of the two numbers is \frac{1}{2}*75 = \frac{75}{2}. 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>

$$r = \frac{75}{2} - u s = \frac{75}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 108

$$(\frac{75}{2} - u) (\frac{75}{2} + u) = 108$$

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

$$\frac{5625}{4} - u^2 = 108$$

Simplify the expression by subtracting \frac{5625}{4} on both sides

$$-u^2 = 108-\frac{5625}{4} = -\frac{5193}{4}$$

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

$$u^2 = \frac{5193}{4} u = \pm\sqrt{\frac{5193}{4}} = \pm \frac{\sqrt{5193}}{2}$$

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

$$r =\frac{75}{2} - \frac{\sqrt{5193}}{2} = 1.469 s = \frac{75}{2} + \frac{\sqrt{5193}}{2} = 73.531$$

Evaluate

$$d^{2}-75d+108$$