Solve 2n^2+13n+15

$$2 n ^ { 2 } + 13 n + 15$$

Factor

$$\left(n+5\right)\left(2n+3\right)$$

Solution Steps

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

$$a+b=13 ab=2\times 15=30$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 30.

$$1,30 2,15 3,10 5,6$$

Calculate the sum for each pair.

$$1+30=31 2+15=17 3+10=13 5+6=11$$

The solution is the pair that gives sum 13.

$$a=3 b=10$$

Rewrite 2n^{2}+13n+15 as \left(2n^{2}+3n\right)+\left(10n+15\right).

$$\left(2n^{2}+3n\right)+\left(10n+15\right)$$

Factor out n in the first and 5 in the second group.

$$n\left(2n+3\right)+5\left(2n+3\right)$$

Factor out common term 2n+3 by using distributive property.

$$\left(2n+3\right)\left(n+5\right)$$

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.

$$2n^{2}+13n+15=0$$

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.

$$n=\frac{-13±\sqrt{13^{2}-4\times 2\times 15}}{2\times 2}$$

Square 13.

$$n=\frac{-13±\sqrt{169-4\times 2\times 15}}{2\times 2}$$

Multiply -4 times 2.

$$n=\frac{-13±\sqrt{169-8\times 15}}{2\times 2}$$

Multiply -8 times 15.

$$n=\frac{-13±\sqrt{169-120}}{2\times 2}$$

Add 169 to -120.

$$n=\frac{-13±\sqrt{49}}{2\times 2}$$

Take the square root of 49.

$$n=\frac{-13±7}{2\times 2}$$

Multiply 2 times 2.

$$n=\frac{-13±7}{4}$$

Now solve the equation n=\frac{-13±7}{4} when ± is plus. Add -13 to 7.

$$n=-\frac{6}{4}$$

Reduce the fraction \frac{-6}{4} to lowest terms by extracting and canceling out 2.

$$n=-\frac{3}{2}$$

Now solve the equation n=\frac{-13±7}{4} when ± is minus. Subtract 7 from -13.

$$n=-\frac{20}{4}$$

Divide -20 by 4.

$$n=-5$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{3}{2} for x_{1} and -5 for x_{2}.

$$2n^{2}+13n+15=2\left(n-\left(-\frac{3}{2}\right)\right)\left(n-\left(-5\right)\right)$$

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

$$2n^{2}+13n+15=2\left(n+\frac{3}{2}\right)\left(n+5\right)$$

Add \frac{3}{2} to n by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$2n^{2}+13n+15=2\times \frac{2n+3}{2}\left(n+5\right)$$

Cancel out 2, the greatest common factor in 2 and 2.

$$2n^{2}+13n+15=\left(2n+3\right)\left(n+5\right)$$

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 2

$$x ^ 2 +\frac{13}{2}x +\frac{15}{2} = 0$$

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

$$r + s = -\frac{13}{2} rs = \frac{15}{2}$$

Two numbers r and s sum up to -\frac{13}{2} exactly when the average of the two numbers is \frac{1}{2}*-\frac{13}{2} = -\frac{13}{4}. 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>

$$r = -\frac{13}{4} - u s = -\frac{13}{4} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{15}{2}

$$(-\frac{13}{4} - u) (-\frac{13}{4} + u) = \frac{15}{2}$$

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

$$\frac{169}{16} - u^2 = \frac{15}{2}$$

Simplify the expression by subtracting \frac{169}{16} on both sides

$$-u^2 = \frac{15}{2}-\frac{169}{16} = -\frac{49}{16}$$

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

$$u^2 = \frac{49}{16} u = \pm\sqrt{\frac{49}{16}} = \pm \frac{7}{4}$$

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

$$r =-\frac{13}{4} - \frac{7}{4} = -5 s = -\frac{13}{4} + \frac{7}{4} = -1.500$$

Evaluate

$$\left(n+5\right)\left(2n+3\right)$$