Solve 13d^2-73d-98

$$13 d ^ { 2 } - 73 d - 98$$

Factor

$$13\left(d-\frac{73-5\sqrt{417}}{26}\right)\left(d-\frac{5\sqrt{417}+73}{26}\right)$$

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.

$$13d^{2}-73d-98=0$$

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.

$$d=\frac{-\left(-73\right)±\sqrt{\left(-73\right)^{2}-4\times 13\left(-98\right)}}{2\times 13}$$

Square -73.

$$d=\frac{-\left(-73\right)±\sqrt{5329-4\times 13\left(-98\right)}}{2\times 13}$$

Multiply -4 times 13.

$$d=\frac{-\left(-73\right)±\sqrt{5329-52\left(-98\right)}}{2\times 13}$$

Multiply -52 times -98.

$$d=\frac{-\left(-73\right)±\sqrt{5329+5096}}{2\times 13}$$

Add 5329 to 5096.

$$d=\frac{-\left(-73\right)±\sqrt{10425}}{2\times 13}$$

Take the square root of 10425.

$$d=\frac{-\left(-73\right)±5\sqrt{417}}{2\times 13}$$

The opposite of -73 is 73.

$$d=\frac{73±5\sqrt{417}}{2\times 13}$$

Multiply 2 times 13.

$$d=\frac{73±5\sqrt{417}}{26}$$

Now solve the equation d=\frac{73±5\sqrt{417}}{26} when ± is plus. Add 73 to 5\sqrt{417}.

$$d=\frac{5\sqrt{417}+73}{26}$$

Now solve the equation d=\frac{73±5\sqrt{417}}{26} when ± is minus. Subtract 5\sqrt{417} from 73.

$$d=\frac{73-5\sqrt{417}}{26}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{73+5\sqrt{417}}{26} for x_{1} and \frac{73-5\sqrt{417}}{26} for x_{2}.

$$13d^{2}-73d-98=13\left(d-\frac{5\sqrt{417}+73}{26}\right)\left(d-\frac{73-5\sqrt{417}}{26}\right)$$

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.This is achieved by dividing both sides of the equation by 13

$$x ^ 2 -\frac{73}{13}x -\frac{98}{13} = 0$$

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

$$r + s = \frac{73}{13} rs = -\frac{98}{13}$$

Two numbers r and s sum up to \frac{73}{13} exactly when the average of the two numbers is \frac{1}{2}*\frac{73}{13} = \frac{73}{26}. 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>

$$r = \frac{73}{26} - u s = \frac{73}{26} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{98}{13}

$$(\frac{73}{26} - u) (\frac{73}{26} + u) = -\frac{98}{13}$$

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

$$\frac{5329}{676} - u^2 = -\frac{98}{13}$$

Simplify the expression by subtracting \frac{5329}{676} on both sides

$$-u^2 = -\frac{98}{13}-\frac{5329}{676} = -\frac{10425}{676}$$

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

$$u^2 = \frac{10425}{676} u = \pm\sqrt{\frac{10425}{676}} = \pm \frac{\sqrt{10425}}{26}$$

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

$$r =\frac{73}{26} - \frac{\sqrt{10425}}{26} = -1.119 s = \frac{73}{26} + \frac{\sqrt{10425}}{26} = 6.735$$

Evaluate

$$13d^{2}-73d-98$$