Solve 15n^2-15n-90

$$15 n ^ { 2 } - 15 n - 90$$

Factor

$$15\left(n-3\right)\left(n+2\right)$$

Solution Steps

Factor out 15.

$$15\left(n^{2}-n-6\right)$$

Consider n^{2}-n-6. Factor the expression by grouping. First, the expression needs to be rewritten as n^{2}+an+bn-6. To find a and b, set up a system to be solved.

$$a+b=-1 ab=1\left(-6\right)=-6$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -6.

$$1,-6 2,-3$$

Calculate the sum for each pair.

$$1-6=-5 2-3=-1$$

The solution is the pair that gives sum -1.

$$a=-3 b=2$$

Rewrite n^{2}-n-6 as \left(n^{2}-3n\right)+\left(2n-6\right).

$$\left(n^{2}-3n\right)+\left(2n-6\right)$$

Factor out n in the first and 2 in the second group.

$$n\left(n-3\right)+2\left(n-3\right)$$

Factor out common term n-3 by using distributive property.

$$\left(n-3\right)\left(n+2\right)$$

Rewrite the complete factored expression.

$$15\left(n-3\right)\left(n+2\right)$$

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.

$$15n^{2}-15n-90=0$$

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.

$$n=\frac{-\left(-15\right)±\sqrt{\left(-15\right)^{2}-4\times 15\left(-90\right)}}{2\times 15}$$

Square -15.

$$n=\frac{-\left(-15\right)±\sqrt{225-4\times 15\left(-90\right)}}{2\times 15}$$

Multiply -4 times 15.

$$n=\frac{-\left(-15\right)±\sqrt{225-60\left(-90\right)}}{2\times 15}$$

Multiply -60 times -90.

$$n=\frac{-\left(-15\right)±\sqrt{225+5400}}{2\times 15}$$

Add 225 to 5400.

$$n=\frac{-\left(-15\right)±\sqrt{5625}}{2\times 15}$$

Take the square root of 5625.

$$n=\frac{-\left(-15\right)±75}{2\times 15}$$

The opposite of -15 is 15.

$$n=\frac{15±75}{2\times 15}$$

Multiply 2 times 15.

$$n=\frac{15±75}{30}$$

Now solve the equation n=\frac{15±75}{30} when ± is plus. Add 15 to 75.

$$n=\frac{90}{30}$$

Divide 90 by 30.

$$n=3$$

Now solve the equation n=\frac{15±75}{30} when ± is minus. Subtract 75 from 15.

$$n=-\frac{60}{30}$$

Divide -60 by 30.

$$n=-2$$

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

$$15n^{2}-15n-90=15\left(n-3\right)\left(n-\left(-2\right)\right)$$

Simplify all the expressions of the form p-\left(-q\right) to p+q.

$$15n^{2}-15n-90=15\left(n-3\right)\left(n+2\right)$$

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.This is achieved by dividing both sides of the equation by 15

$$x ^ 2 -1x -6 = 0$$

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

$$r + s = 1 rs = -6$$

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

$$r = \frac{1}{2} - u s = \frac{1}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -6

$$(\frac{1}{2} - u) (\frac{1}{2} + u) = -6$$

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

$$\frac{1}{4} - u^2 = -6$$

Simplify the expression by subtracting \frac{1}{4} on both sides

$$-u^2 = -6-\frac{1}{4} = -\frac{25}{4}$$

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

$$u^2 = \frac{25}{4} u = \pm\sqrt{\frac{25}{4}} = \pm \frac{5}{2}$$

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

$$r =\frac{1}{2} - \frac{5}{2} = -2 s = \frac{1}{2} + \frac{5}{2} = 3$$

Evaluate

$$15\left(n-3\right)\left(n+2\right)$$