Solve y^2+59y-24

$$y ^ { 2 } + 59 y - 24$$

Factor

$$\left(y-\frac{-7\sqrt{73}-59}{2}\right)\left(y-\frac{7\sqrt{73}-59}{2}\right)$$

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.

$$y^{2}+59y-24=0$$

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.

$$y=\frac{-59±\sqrt{59^{2}-4\left(-24\right)}}{2}$$

Square 59.

$$y=\frac{-59±\sqrt{3481-4\left(-24\right)}}{2}$$

Multiply -4 times -24.

$$y=\frac{-59±\sqrt{3481+96}}{2}$$

Add 3481 to 96.

$$y=\frac{-59±\sqrt{3577}}{2}$$

Take the square root of 3577.

$$y=\frac{-59±7\sqrt{73}}{2}$$

Now solve the equation y=\frac{-59±7\sqrt{73}}{2} when ± is plus. Add -59 to 7\sqrt{73}.

$$y=\frac{7\sqrt{73}-59}{2}$$

Now solve the equation y=\frac{-59±7\sqrt{73}}{2} when ± is minus. Subtract 7\sqrt{73} from -59.

$$y=\frac{-7\sqrt{73}-59}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-59+7\sqrt{73}}{2} for x_{1} and \frac{-59-7\sqrt{73}}{2} for x_{2}.

$$y^{2}+59y-24=\left(y-\frac{7\sqrt{73}-59}{2}\right)\left(y-\frac{-7\sqrt{73}-59}{2}\right)$$

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.

$$x ^ 2 +59x -24 = 0$$

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

$$r + s = -59 rs = -24$$

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

$$r = -\frac{59}{2} - u s = -\frac{59}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -24

$$(-\frac{59}{2} - u) (-\frac{59}{2} + u) = -24$$

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

$$\frac{3481}{4} - u^2 = -24$$

Simplify the expression by subtracting \frac{3481}{4} on both sides

$$-u^2 = -24-\frac{3481}{4} = -\frac{3577}{4}$$

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

$$u^2 = \frac{3577}{4} u = \pm\sqrt{\frac{3577}{4}} = \pm \frac{\sqrt{3577}}{2}$$

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

$$r =-\frac{59}{2} - \frac{\sqrt{3577}}{2} = -59.404 s = -\frac{59}{2} + \frac{\sqrt{3577}}{2} = 0.404$$

Evaluate

$$y^{2}+59y-24$$