Solve 5x^2-36x-81

$$5 x ^ { 2 } - 36 x - 81$$

Factor

$$\left(x-9\right)\left(5x+9\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 5x^{2}+ax+bx-81. To find a and b, set up a system to be solved.

$$a+b=-36 ab=5\left(-81\right)=-405$$

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 -405.

$$1,-405 3,-135 5,-81 9,-45 15,-27$$

Calculate the sum for each pair.

$$1-405=-404 3-135=-132 5-81=-76 9-45=-36 15-27=-12$$

The solution is the pair that gives sum -36.

$$a=-45 b=9$$

Rewrite 5x^{2}-36x-81 as \left(5x^{2}-45x\right)+\left(9x-81\right).

$$\left(5x^{2}-45x\right)+\left(9x-81\right)$$

Factor out 5x in the first and 9 in the second group.

$$5x\left(x-9\right)+9\left(x-9\right)$$

Factor out common term x-9 by using distributive property.

$$\left(x-9\right)\left(5x+9\right)$$

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.

$$5x^{2}-36x-81=0$$

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.

$$x=\frac{-\left(-36\right)±\sqrt{\left(-36\right)^{2}-4\times 5\left(-81\right)}}{2\times 5}$$

Square -36.

$$x=\frac{-\left(-36\right)±\sqrt{1296-4\times 5\left(-81\right)}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-\left(-36\right)±\sqrt{1296-20\left(-81\right)}}{2\times 5}$$

Multiply -20 times -81.

$$x=\frac{-\left(-36\right)±\sqrt{1296+1620}}{2\times 5}$$

Add 1296 to 1620.

$$x=\frac{-\left(-36\right)±\sqrt{2916}}{2\times 5}$$

Take the square root of 2916.

$$x=\frac{-\left(-36\right)±54}{2\times 5}$$

The opposite of -36 is 36.

$$x=\frac{36±54}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{36±54}{10}$$

Now solve the equation x=\frac{36±54}{10} when ± is plus. Add 36 to 54.

$$x=\frac{90}{10}$$

Divide 90 by 10.

$$x=9$$

Now solve the equation x=\frac{36±54}{10} when ± is minus. Subtract 54 from 36.

$$x=-\frac{18}{10}$$

Reduce the fraction \frac{-18}{10} to lowest terms by extracting and canceling out 2.

$$x=-\frac{9}{5}$$

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

$$5x^{2}-36x-81=5\left(x-9\right)\left(x-\left(-\frac{9}{5}\right)\right)$$

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

$$5x^{2}-36x-81=5\left(x-9\right)\left(x+\frac{9}{5}\right)$$

Add \frac{9}{5} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$5x^{2}-36x-81=5\left(x-9\right)\times \frac{5x+9}{5}$$

Cancel out 5, the greatest common factor in 5 and 5.

$$5x^{2}-36x-81=\left(x-9\right)\left(5x+9\right)$$

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 5

$$x ^ 2 -\frac{36}{5}x -\frac{81}{5} = 0$$

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

$$r + s = \frac{36}{5} rs = -\frac{81}{5}$$

Two numbers r and s sum up to \frac{36}{5} exactly when the average of the two numbers is \frac{1}{2}*\frac{36}{5} = \frac{18}{5}. 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>

$$r = \frac{18}{5} - u s = \frac{18}{5} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{81}{5}

$$(\frac{18}{5} - u) (\frac{18}{5} + u) = -\frac{81}{5}$$

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

$$\frac{324}{25} - u^2 = -\frac{81}{5}$$

Simplify the expression by subtracting \frac{324}{25} on both sides

$$-u^2 = -\frac{81}{5}-\frac{324}{25} = -\frac{729}{25}$$

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

$$u^2 = \frac{729}{25} u = \pm\sqrt{\frac{729}{25}} = \pm \frac{27}{5}$$

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

$$r =\frac{18}{5} - \frac{27}{5} = -1.800 s = \frac{18}{5} + \frac{27}{5} = 9$$

Evaluate

$$\left(x-9\right)\left(5x+9\right)$$