Solve 9x^2-59x-120

$$9 x ^ { 2 } - 59 x - 120$$

Factor

$$9\left(x-\frac{59-\sqrt{7801}}{18}\right)\left(x-\frac{\sqrt{7801}+59}{18}\right)$$

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.

$$9x^{2}-59x-120=0$$

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.

$$x=\frac{-\left(-59\right)±\sqrt{\left(-59\right)^{2}-4\times 9\left(-120\right)}}{2\times 9}$$

Square -59.

$$x=\frac{-\left(-59\right)±\sqrt{3481-4\times 9\left(-120\right)}}{2\times 9}$$

Multiply -4 times 9.

$$x=\frac{-\left(-59\right)±\sqrt{3481-36\left(-120\right)}}{2\times 9}$$

Multiply -36 times -120.

$$x=\frac{-\left(-59\right)±\sqrt{3481+4320}}{2\times 9}$$

Add 3481 to 4320.

$$x=\frac{-\left(-59\right)±\sqrt{7801}}{2\times 9}$$

The opposite of -59 is 59.

$$x=\frac{59±\sqrt{7801}}{2\times 9}$$

Multiply 2 times 9.

$$x=\frac{59±\sqrt{7801}}{18}$$

Now solve the equation x=\frac{59±\sqrt{7801}}{18} when ± is plus. Add 59 to \sqrt{7801}.

$$x=\frac{\sqrt{7801}+59}{18}$$

Now solve the equation x=\frac{59±\sqrt{7801}}{18} when ± is minus. Subtract \sqrt{7801} from 59.

$$x=\frac{59-\sqrt{7801}}{18}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{59+\sqrt{7801}}{18} for x_{1} and \frac{59-\sqrt{7801}}{18} for x_{2}.

$$9x^{2}-59x-120=9\left(x-\frac{\sqrt{7801}+59}{18}\right)\left(x-\frac{59-\sqrt{7801}}{18}\right)$$

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 9

$$x ^ 2 -\frac{59}{9}x -\frac{40}{3} = 0$$

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

$$r + s = \frac{59}{9} rs = -\frac{40}{3}$$

Two numbers r and s sum up to \frac{59}{9} exactly when the average of the two numbers is \frac{1}{2}*\frac{59}{9} = \frac{59}{18}. 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>

$$r = \frac{59}{18} - u s = \frac{59}{18} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{40}{3}

$$(\frac{59}{18} - u) (\frac{59}{18} + u) = -\frac{40}{3}$$

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

$$\frac{3481}{324} - u^2 = -\frac{40}{3}$$

Simplify the expression by subtracting \frac{3481}{324} on both sides

$$-u^2 = -\frac{40}{3}-\frac{3481}{324} = -\frac{7801}{324}$$

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

$$u^2 = \frac{7801}{324} u = \pm\sqrt{\frac{7801}{324}} = \pm \frac{\sqrt{7801}}{18}$$

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

$$r =\frac{59}{18} - \frac{\sqrt{7801}}{18} = -1.629 s = \frac{59}{18} + \frac{\sqrt{7801}}{18} = 8.185$$

Evaluate

$$9x^{2}-59x-120$$