Solve 13t^2+69t-602

$$13 t ^ { 2 } + 69 t - 602$$

Factor

$$13\left(t-\frac{-\sqrt{36065}-69}{26}\right)\left(t-\frac{\sqrt{36065}-69}{26}\right)$$

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.

$$13t^{2}+69t-602=0$$

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.

$$t=\frac{-69±\sqrt{69^{2}-4\times 13\left(-602\right)}}{2\times 13}$$

Square 69.

$$t=\frac{-69±\sqrt{4761-4\times 13\left(-602\right)}}{2\times 13}$$

Multiply -4 times 13.

$$t=\frac{-69±\sqrt{4761-52\left(-602\right)}}{2\times 13}$$

Multiply -52 times -602.

$$t=\frac{-69±\sqrt{4761+31304}}{2\times 13}$$

Add 4761 to 31304.

$$t=\frac{-69±\sqrt{36065}}{2\times 13}$$

Multiply 2 times 13.

$$t=\frac{-69±\sqrt{36065}}{26}$$

Now solve the equation t=\frac{-69±\sqrt{36065}}{26} when ± is plus. Add -69 to \sqrt{36065}.

$$t=\frac{\sqrt{36065}-69}{26}$$

Now solve the equation t=\frac{-69±\sqrt{36065}}{26} when ± is minus. Subtract \sqrt{36065} from -69.

$$t=\frac{-\sqrt{36065}-69}{26}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-69+\sqrt{36065}}{26} for x_{1} and \frac{-69-\sqrt{36065}}{26} for x_{2}.

$$13t^{2}+69t-602=13\left(t-\frac{\sqrt{36065}-69}{26}\right)\left(t-\frac{-\sqrt{36065}-69}{26}\right)$$

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

$$x ^ 2 +\frac{69}{13}x -\frac{602}{13} = 0$$

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

$$r + s = -\frac{69}{13} rs = -\frac{602}{13}$$

Two numbers r and s sum up to -\frac{69}{13} exactly when the average of the two numbers is \frac{1}{2}*-\frac{69}{13} = -\frac{69}{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>

$$r = -\frac{69}{26} - u s = -\frac{69}{26} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{602}{13}

$$(-\frac{69}{26} - u) (-\frac{69}{26} + u) = -\frac{602}{13}$$

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

$$\frac{4761}{676} - u^2 = -\frac{602}{13}$$

Simplify the expression by subtracting \frac{4761}{676} on both sides

$$-u^2 = -\frac{602}{13}-\frac{4761}{676} = -\frac{36065}{676}$$

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

$$u^2 = \frac{36065}{676} u = \pm\sqrt{\frac{36065}{676}} = \pm \frac{\sqrt{36065}}{26}$$

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

$$r =-\frac{69}{26} - \frac{\sqrt{36065}}{26} = -9.958 s = -\frac{69}{26} + \frac{\sqrt{36065}}{26} = 4.650$$

Evaluate

$$13t^{2}+69t-602$$