Solve 5n^2+74n-15

$$5 n ^ { 2 } + 74 n - 15$$

Factor

$$\left(5n-1\right)\left(n+15\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 5n^{2}+an+bn-15. To find a and b, set up a system to be solved.

$$a+b=74 ab=5\left(-15\right)=-75$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -75.

$$-1,75 -3,25 -5,15$$

Calculate the sum for each pair.

$$-1+75=74 -3+25=22 -5+15=10$$

The solution is the pair that gives sum 74.

$$a=-1 b=75$$

Rewrite 5n^{2}+74n-15 as \left(5n^{2}-n\right)+\left(75n-15\right).

$$\left(5n^{2}-n\right)+\left(75n-15\right)$$

Factor out n in the first and 15 in the second group.

$$n\left(5n-1\right)+15\left(5n-1\right)$$

Factor out common term 5n-1 by using distributive property.

$$\left(5n-1\right)\left(n+15\right)$$

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.

$$5n^{2}+74n-15=0$$

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.

$$n=\frac{-74±\sqrt{74^{2}-4\times 5\left(-15\right)}}{2\times 5}$$

Square 74.

$$n=\frac{-74±\sqrt{5476-4\times 5\left(-15\right)}}{2\times 5}$$

Multiply -4 times 5.

$$n=\frac{-74±\sqrt{5476-20\left(-15\right)}}{2\times 5}$$

Multiply -20 times -15.

$$n=\frac{-74±\sqrt{5476+300}}{2\times 5}$$

Add 5476 to 300.

$$n=\frac{-74±\sqrt{5776}}{2\times 5}$$

Take the square root of 5776.

$$n=\frac{-74±76}{2\times 5}$$

Multiply 2 times 5.

$$n=\frac{-74±76}{10}$$

Now solve the equation n=\frac{-74±76}{10} when ± is plus. Add -74 to 76.

$$n=\frac{2}{10}$$

Reduce the fraction \frac{2}{10} to lowest terms by extracting and canceling out 2.

$$n=\frac{1}{5}$$

Now solve the equation n=\frac{-74±76}{10} when ± is minus. Subtract 76 from -74.

$$n=-\frac{150}{10}$$

Divide -150 by 10.

$$n=-15$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{1}{5} for x_{1} and -15 for x_{2}.

$$5n^{2}+74n-15=5\left(n-\frac{1}{5}\right)\left(n-\left(-15\right)\right)$$

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

$$5n^{2}+74n-15=5\left(n-\frac{1}{5}\right)\left(n+15\right)$$

Subtract \frac{1}{5} from n by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$5n^{2}+74n-15=5\times \frac{5n-1}{5}\left(n+15\right)$$

Cancel out 5, the greatest common factor in 5 and 5.

$$5n^{2}+74n-15=\left(5n-1\right)\left(n+15\right)$$

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 5

$$x ^ 2 +\frac{74}{5}x -3 = 0$$

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

$$r + s = -\frac{74}{5} rs = -3$$

Two numbers r and s sum up to -\frac{74}{5} exactly when the average of the two numbers is \frac{1}{2}*-\frac{74}{5} = -\frac{37}{5}. 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-gzdabgg4ehffg0hf.b01.azurefd.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = -\frac{37}{5} - u s = -\frac{37}{5} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -3

$$(-\frac{37}{5} - u) (-\frac{37}{5} + u) = -3$$

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

$$\frac{1369}{25} - u^2 = -3$$

Simplify the expression by subtracting \frac{1369}{25} on both sides

$$-u^2 = -3-\frac{1369}{25} = -\frac{1444}{25}$$

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

$$u^2 = \frac{1444}{25} u = \pm\sqrt{\frac{1444}{25}} = \pm \frac{38}{5}$$

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

$$r =-\frac{37}{5} - \frac{38}{5} = -15 s = -\frac{37}{5} + \frac{38}{5} = 0.200$$

Evaluate

$$\left(5n-1\right)\left(n+15\right)$$