Solve 24n^2+231n+135

$$24 n ^ { 2 } + 231 n + 135$$

Factor

$$3\left(n+9\right)\left(8n+5\right)$$

Solution Steps

Factor out 3.

$$3\left(8n^{2}+77n+45\right)$$

Consider 8n^{2}+77n+45. Factor the expression by grouping. First, the expression needs to be rewritten as 8n^{2}+an+bn+45. To find a and b, set up a system to be solved.

$$a+b=77 ab=8\times 45=360$$

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 360.

$$1,360 2,180 3,120 4,90 5,72 6,60 8,45 9,40 10,36 12,30 15,24 18,20$$

Calculate the sum for each pair.

$$1+360=361 2+180=182 3+120=123 4+90=94 5+72=77 6+60=66 8+45=53 9+40=49 10+36=46 12+30=42 15+24=39 18+20=38$$

The solution is the pair that gives sum 77.

$$a=5 b=72$$

Rewrite 8n^{2}+77n+45 as \left(8n^{2}+5n\right)+\left(72n+45\right).

$$\left(8n^{2}+5n\right)+\left(72n+45\right)$$

Factor out n in the first and 9 in the second group.

$$n\left(8n+5\right)+9\left(8n+5\right)$$

Factor out common term 8n+5 by using distributive property.

$$\left(8n+5\right)\left(n+9\right)$$

Rewrite the complete factored expression.

$$3\left(8n+5\right)\left(n+9\right)$$

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.

$$24n^{2}+231n+135=0$$

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.

$$n=\frac{-231±\sqrt{231^{2}-4\times 24\times 135}}{2\times 24}$$

Square 231.

$$n=\frac{-231±\sqrt{53361-4\times 24\times 135}}{2\times 24}$$

Multiply -4 times 24.

$$n=\frac{-231±\sqrt{53361-96\times 135}}{2\times 24}$$

Multiply -96 times 135.

$$n=\frac{-231±\sqrt{53361-12960}}{2\times 24}$$

Add 53361 to -12960.

$$n=\frac{-231±\sqrt{40401}}{2\times 24}$$

Take the square root of 40401.

$$n=\frac{-231±201}{2\times 24}$$

Multiply 2 times 24.

$$n=\frac{-231±201}{48}$$

Now solve the equation n=\frac{-231±201}{48} when ± is plus. Add -231 to 201.

$$n=-\frac{30}{48}$$

Reduce the fraction \frac{-30}{48} to lowest terms by extracting and canceling out 6.

$$n=-\frac{5}{8}$$

Now solve the equation n=\frac{-231±201}{48} when ± is minus. Subtract 201 from -231.

$$n=-\frac{432}{48}$$

Divide -432 by 48.

$$n=-9$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{5}{8} for x_{1} and -9 for x_{2}.

$$24n^{2}+231n+135=24\left(n-\left(-\frac{5}{8}\right)\right)\left(n-\left(-9\right)\right)$$

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

$$24n^{2}+231n+135=24\left(n+\frac{5}{8}\right)\left(n+9\right)$$

Add \frac{5}{8} to n by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$24n^{2}+231n+135=24\times \frac{8n+5}{8}\left(n+9\right)$$

Cancel out 8, the greatest common factor in 24 and 8.

$$24n^{2}+231n+135=3\left(8n+5\right)\left(n+9\right)$$

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 24

$$x ^ 2 +\frac{77}{8}x +\frac{45}{8} = 0$$

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

$$r + s = -\frac{77}{8} rs = \frac{45}{8}$$

Two numbers r and s sum up to -\frac{77}{8} exactly when the average of the two numbers is \frac{1}{2}*-\frac{77}{8} = -\frac{77}{16}. 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>

$$r = -\frac{77}{16} - u s = -\frac{77}{16} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{45}{8}

$$(-\frac{77}{16} - u) (-\frac{77}{16} + u) = \frac{45}{8}$$

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

$$\frac{5929}{256} - u^2 = \frac{45}{8}$$

Simplify the expression by subtracting \frac{5929}{256} on both sides

$$-u^2 = \frac{45}{8}-\frac{5929}{256} = -\frac{4489}{256}$$

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

$$u^2 = \frac{4489}{256} u = \pm\sqrt{\frac{4489}{256}} = \pm \frac{67}{16}$$

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

$$r =-\frac{77}{16} - \frac{67}{16} = -9 s = -\frac{77}{16} + \frac{67}{16} = -0.625$$

Evaluate

$$3\left(n+9\right)\left(8n+5\right)$$