Solve 6p^2+37p+45

$$6 p ^ { 2 } + 37 p + 45$$

Factor

$$\left(2p+9\right)\left(3p+5\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 6p^{2}+ap+bp+45. To find a and b, set up a system to be solved.

$$a+b=37 ab=6\times 45=270$$

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 270.

$$1,270 2,135 3,90 5,54 6,45 9,30 10,27 15,18$$

Calculate the sum for each pair.

$$1+270=271 2+135=137 3+90=93 5+54=59 6+45=51 9+30=39 10+27=37 15+18=33$$

The solution is the pair that gives sum 37.

$$a=10 b=27$$

Rewrite 6p^{2}+37p+45 as \left(6p^{2}+10p\right)+\left(27p+45\right).

$$\left(6p^{2}+10p\right)+\left(27p+45\right)$$

Factor out 2p in the first and 9 in the second group.

$$2p\left(3p+5\right)+9\left(3p+5\right)$$

Factor out common term 3p+5 by using distributive property.

$$\left(3p+5\right)\left(2p+9\right)$$

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.

$$6p^{2}+37p+45=0$$

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.

$$p=\frac{-37±\sqrt{37^{2}-4\times 6\times 45}}{2\times 6}$$

Square 37.

$$p=\frac{-37±\sqrt{1369-4\times 6\times 45}}{2\times 6}$$

Multiply -4 times 6.

$$p=\frac{-37±\sqrt{1369-24\times 45}}{2\times 6}$$

Multiply -24 times 45.

$$p=\frac{-37±\sqrt{1369-1080}}{2\times 6}$$

Add 1369 to -1080.

$$p=\frac{-37±\sqrt{289}}{2\times 6}$$

Take the square root of 289.

$$p=\frac{-37±17}{2\times 6}$$

Multiply 2 times 6.

$$p=\frac{-37±17}{12}$$

Now solve the equation p=\frac{-37±17}{12} when ± is plus. Add -37 to 17.

$$p=-\frac{20}{12}$$

Reduce the fraction \frac{-20}{12} to lowest terms by extracting and canceling out 4.

$$p=-\frac{5}{3}$$

Now solve the equation p=\frac{-37±17}{12} when ± is minus. Subtract 17 from -37.

$$p=-\frac{54}{12}$$

Reduce the fraction \frac{-54}{12} to lowest terms by extracting and canceling out 6.

$$p=-\frac{9}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{5}{3} for x_{1} and -\frac{9}{2} for x_{2}.

$$6p^{2}+37p+45=6\left(p-\left(-\frac{5}{3}\right)\right)\left(p-\left(-\frac{9}{2}\right)\right)$$

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

$$6p^{2}+37p+45=6\left(p+\frac{5}{3}\right)\left(p+\frac{9}{2}\right)$$

Add \frac{5}{3} to p by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$6p^{2}+37p+45=6\times \frac{3p+5}{3}\left(p+\frac{9}{2}\right)$$

Add \frac{9}{2} to p by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$6p^{2}+37p+45=6\times \frac{3p+5}{3}\times \frac{2p+9}{2}$$

Multiply \frac{3p+5}{3} times \frac{2p+9}{2} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$6p^{2}+37p+45=6\times \frac{\left(3p+5\right)\left(2p+9\right)}{3\times 2}$$

Multiply 3 times 2.

$$6p^{2}+37p+45=6\times \frac{\left(3p+5\right)\left(2p+9\right)}{6}$$

Cancel out 6, the greatest common factor in 6 and 6.

$$6p^{2}+37p+45=\left(3p+5\right)\left(2p+9\right)$$

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 6

$$x ^ 2 +\frac{37}{6}x +\frac{15}{2} = 0$$

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

$$r + s = -\frac{37}{6} rs = \frac{15}{2}$$

Two numbers r and s sum up to -\frac{37}{6} exactly when the average of the two numbers is \frac{1}{2}*-\frac{37}{6} = -\frac{37}{12}. 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>

$$r = -\frac{37}{12} - u s = -\frac{37}{12} + u$$

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

$$(-\frac{37}{12} - u) (-\frac{37}{12} + u) = \frac{15}{2}$$

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

$$\frac{1369}{144} - u^2 = \frac{15}{2}$$

Simplify the expression by subtracting \frac{1369}{144} on both sides

$$-u^2 = \frac{15}{2}-\frac{1369}{144} = -\frac{289}{144}$$

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

$$u^2 = \frac{289}{144} u = \pm\sqrt{\frac{289}{144}} = \pm \frac{17}{12}$$

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

$$r =-\frac{37}{12} - \frac{17}{12} = -4.500 s = -\frac{37}{12} + \frac{17}{12} = -1.667$$

Evaluate

$$\left(2p+9\right)\left(3p+5\right)$$