Solve 15n^2+45n-50

$$15 n ^ { 2 } + 45 n - 50$$

Factor

$$15\left(n-\left(-\frac{\sqrt{201}}{6}-\frac{3}{2}\right)\right)\left(n-\left(\frac{\sqrt{201}}{6}-\frac{3}{2}\right)\right)$$

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.

$$15n^{2}+45n-50=0$$

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.

$$n=\frac{-45±\sqrt{45^{2}-4\times 15\left(-50\right)}}{2\times 15}$$

Square 45.

$$n=\frac{-45±\sqrt{2025-4\times 15\left(-50\right)}}{2\times 15}$$

Multiply -4 times 15.

$$n=\frac{-45±\sqrt{2025-60\left(-50\right)}}{2\times 15}$$

Multiply -60 times -50.

$$n=\frac{-45±\sqrt{2025+3000}}{2\times 15}$$

Add 2025 to 3000.

$$n=\frac{-45±\sqrt{5025}}{2\times 15}$$

Take the square root of 5025.

$$n=\frac{-45±5\sqrt{201}}{2\times 15}$$

Multiply 2 times 15.

$$n=\frac{-45±5\sqrt{201}}{30}$$

Now solve the equation n=\frac{-45±5\sqrt{201}}{30} when ± is plus. Add -45 to 5\sqrt{201}.

$$n=\frac{5\sqrt{201}-45}{30}$$

Divide -45+5\sqrt{201} by 30.

$$n=\frac{\sqrt{201}}{6}-\frac{3}{2}$$

Now solve the equation n=\frac{-45±5\sqrt{201}}{30} when ± is minus. Subtract 5\sqrt{201} from -45.

$$n=\frac{-5\sqrt{201}-45}{30}$$

Divide -45-5\sqrt{201} by 30.

$$n=-\frac{\sqrt{201}}{6}-\frac{3}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{3}{2}+\frac{\sqrt{201}}{6} for x_{1} and -\frac{3}{2}-\frac{\sqrt{201}}{6} for x_{2}.

$$15n^{2}+45n-50=15\left(n-\left(\frac{\sqrt{201}}{6}-\frac{3}{2}\right)\right)\left(n-\left(-\frac{\sqrt{201}}{6}-\frac{3}{2}\right)\right)$$

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 15

$$x ^ 2 +3x -\frac{10}{3} = 0$$

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

$$r + s = -3 rs = -\frac{10}{3}$$

Two numbers r and s sum up to -3 exactly when the average of the two numbers is \frac{1}{2}*-3 = -\frac{3}{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>

$$r = -\frac{3}{2} - u s = -\frac{3}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{10}{3}

$$(-\frac{3}{2} - u) (-\frac{3}{2} + u) = -\frac{10}{3}$$

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

$$\frac{9}{4} - u^2 = -\frac{10}{3}$$

Simplify the expression by subtracting \frac{9}{4} on both sides

$$-u^2 = -\frac{10}{3}-\frac{9}{4} = -\frac{67}{12}$$

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

$$u^2 = \frac{67}{12} u = \pm\sqrt{\frac{67}{12}} = \pm \frac{\sqrt{67}}{\sqrt{12}}$$

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

$$r =-\frac{3}{2} - \frac{\sqrt{67}}{\sqrt{12}} = -3.863 s = -\frac{3}{2} + \frac{\sqrt{67}}{\sqrt{12}} = 0.863$$

Evaluate

$$15n^{2}+45n-50$$