Solve 3p^2+25p-18

$$3 p ^ { 2 } + 25 p - 18$$

Factor

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

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 3p^{2}+ap+bp-18. To find a and b, set up a system to be solved.

$$a+b=25 ab=3\left(-18\right)=-54$$

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 -54.

$$-1,54 -2,27 -3,18 -6,9$$

Calculate the sum for each pair.

$$-1+54=53 -2+27=25 -3+18=15 -6+9=3$$

The solution is the pair that gives sum 25.

$$a=-2 b=27$$

Rewrite 3p^{2}+25p-18 as \left(3p^{2}-2p\right)+\left(27p-18\right).

$$\left(3p^{2}-2p\right)+\left(27p-18\right)$$

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

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

Factor out common term 3p-2 by using distributive property.

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

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.

$$3p^{2}+25p-18=0$$

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.

$$p=\frac{-25±\sqrt{25^{2}-4\times 3\left(-18\right)}}{2\times 3}$$

Square 25.

$$p=\frac{-25±\sqrt{625-4\times 3\left(-18\right)}}{2\times 3}$$

Multiply -4 times 3.

$$p=\frac{-25±\sqrt{625-12\left(-18\right)}}{2\times 3}$$

Multiply -12 times -18.

$$p=\frac{-25±\sqrt{625+216}}{2\times 3}$$

Add 625 to 216.

$$p=\frac{-25±\sqrt{841}}{2\times 3}$$

Take the square root of 841.

$$p=\frac{-25±29}{2\times 3}$$

Multiply 2 times 3.

$$p=\frac{-25±29}{6}$$

Now solve the equation p=\frac{-25±29}{6} when ± is plus. Add -25 to 29.

$$p=\frac{4}{6}$$

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

$$p=\frac{2}{3}$$

Now solve the equation p=\frac{-25±29}{6} when ± is minus. Subtract 29 from -25.

$$p=-\frac{54}{6}$$

Divide -54 by 6.

$$p=-9$$

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

$$3p^{2}+25p-18=3\left(p-\frac{2}{3}\right)\left(p-\left(-9\right)\right)$$

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

$$3p^{2}+25p-18=3\left(p-\frac{2}{3}\right)\left(p+9\right)$$

Subtract \frac{2}{3} from p by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$3p^{2}+25p-18=3\times \frac{3p-2}{3}\left(p+9\right)$$

Cancel out 3, the greatest common factor in 3 and 3.

$$3p^{2}+25p-18=\left(3p-2\right)\left(p+9\right)$$

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 3

$$x ^ 2 +\frac{25}{3}x -6 = 0$$

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

$$r + s = -\frac{25}{3} rs = -6$$

Two numbers r and s sum up to -\frac{25}{3} exactly when the average of the two numbers is \frac{1}{2}*-\frac{25}{3} = -\frac{25}{6}. 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>

$$r = -\frac{25}{6} - u s = -\frac{25}{6} + u$$

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

$$(-\frac{25}{6} - u) (-\frac{25}{6} + u) = -6$$

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

$$\frac{625}{36} - u^2 = -6$$

Simplify the expression by subtracting \frac{625}{36} on both sides

$$-u^2 = -6-\frac{625}{36} = -\frac{841}{36}$$

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

$$u^2 = \frac{841}{36} u = \pm\sqrt{\frac{841}{36}} = \pm \frac{29}{6}$$

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

$$r =-\frac{25}{6} - \frac{29}{6} = -9 s = -\frac{25}{6} + \frac{29}{6} = 0.667$$

Evaluate

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