Solve 9p^2-8p-1

$$9 p ^ { 2 } - 8 p - 1$$

Factor

$$\left(p-1\right)\left(9p+1\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 9p^{2}+ap+bp-1. To find a and b, set up a system to be solved.

$$a+b=-8 ab=9\left(-1\right)=-9$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -9.

$$1,-9 3,-3$$

Calculate the sum for each pair.

$$1-9=-8 3-3=0$$

The solution is the pair that gives sum -8.

$$a=-9 b=1$$

Rewrite 9p^{2}-8p-1 as \left(9p^{2}-9p\right)+\left(p-1\right).

$$\left(9p^{2}-9p\right)+\left(p-1\right)$$

Factor out 9p in 9p^{2}-9p.

$$9p\left(p-1\right)+p-1$$

Factor out common term p-1 by using distributive property.

$$\left(p-1\right)\left(9p+1\right)$$

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.

$$9p^{2}-8p-1=0$$

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.

$$p=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 9\left(-1\right)}}{2\times 9}$$

Square -8.

$$p=\frac{-\left(-8\right)±\sqrt{64-4\times 9\left(-1\right)}}{2\times 9}$$

Multiply -4 times 9.

$$p=\frac{-\left(-8\right)±\sqrt{64-36\left(-1\right)}}{2\times 9}$$

Multiply -36 times -1.

$$p=\frac{-\left(-8\right)±\sqrt{64+36}}{2\times 9}$$

Add 64 to 36.

$$p=\frac{-\left(-8\right)±\sqrt{100}}{2\times 9}$$

Take the square root of 100.

$$p=\frac{-\left(-8\right)±10}{2\times 9}$$

The opposite of -8 is 8.

$$p=\frac{8±10}{2\times 9}$$

Multiply 2 times 9.

$$p=\frac{8±10}{18}$$

Now solve the equation p=\frac{8±10}{18} when ± is plus. Add 8 to 10.

$$p=\frac{18}{18}$$

Divide 18 by 18.

$$p=1$$

Now solve the equation p=\frac{8±10}{18} when ± is minus. Subtract 10 from 8.

$$p=-\frac{2}{18}$$

Reduce the fraction \frac{-2}{18} to lowest terms by extracting and canceling out 2.

$$p=-\frac{1}{9}$$

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

$$9p^{2}-8p-1=9\left(p-1\right)\left(p-\left(-\frac{1}{9}\right)\right)$$

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

$$9p^{2}-8p-1=9\left(p-1\right)\left(p+\frac{1}{9}\right)$$

Add \frac{1}{9} to p by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$9p^{2}-8p-1=9\left(p-1\right)\times \frac{9p+1}{9}$$

Cancel out 9, the greatest common factor in 9 and 9.

$$9p^{2}-8p-1=\left(p-1\right)\left(9p+1\right)$$

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 9

$$x ^ 2 -\frac{8}{9}x -\frac{1}{9} = 0$$

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

$$r + s = \frac{8}{9} rs = -\frac{1}{9}$$

Two numbers r and s sum up to \frac{8}{9} exactly when the average of the two numbers is \frac{1}{2}*\frac{8}{9} = \frac{4}{9}. 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>

$$r = \frac{4}{9} - u s = \frac{4}{9} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{1}{9}

$$(\frac{4}{9} - u) (\frac{4}{9} + u) = -\frac{1}{9}$$

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

$$\frac{16}{81} - u^2 = -\frac{1}{9}$$

Simplify the expression by subtracting \frac{16}{81} on both sides

$$-u^2 = -\frac{1}{9}-\frac{16}{81} = -\frac{25}{81}$$

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

$$u^2 = \frac{25}{81} u = \pm\sqrt{\frac{25}{81}} = \pm \frac{5}{9}$$

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

$$r =\frac{4}{9} - \frac{5}{9} = -0.111 s = \frac{4}{9} + \frac{5}{9} = 1$$

Evaluate

$$\left(p-1\right)\left(9p+1\right)$$