Solve 9x^2-155x-500

$$9 x ^ { 2 } - 155 x - 500$$

Factor

$$\left(x-20\right)\left(9x+25\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 9x^{2}+ax+bx-500. To find a and b, set up a system to be solved.

$$a+b=-155 ab=9\left(-500\right)=-4500$$

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 -4500.

$$1,-4500 2,-2250 3,-1500 4,-1125 5,-900 6,-750 9,-500 10,-450 12,-375 15,-300 18,-250 20,-225 25,-180 30,-150 36,-125 45,-100 50,-90 60,-75$$

Calculate the sum for each pair.

$$1-4500=-4499 2-2250=-2248 3-1500=-1497 4-1125=-1121 5-900=-895 6-750=-744 9-500=-491 10-450=-440 12-375=-363 15-300=-285 18-250=-232 20-225=-205 25-180=-155 30-150=-120 36-125=-89 45-100=-55 50-90=-40 60-75=-15$$

The solution is the pair that gives sum -155.

$$a=-180 b=25$$

Rewrite 9x^{2}-155x-500 as \left(9x^{2}-180x\right)+\left(25x-500\right).

$$\left(9x^{2}-180x\right)+\left(25x-500\right)$$

Factor out 9x in the first and 25 in the second group.

$$9x\left(x-20\right)+25\left(x-20\right)$$

Factor out common term x-20 by using distributive property.

$$\left(x-20\right)\left(9x+25\right)$$

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.

$$9x^{2}-155x-500=0$$

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.

$$x=\frac{-\left(-155\right)±\sqrt{\left(-155\right)^{2}-4\times 9\left(-500\right)}}{2\times 9}$$

Square -155.

$$x=\frac{-\left(-155\right)±\sqrt{24025-4\times 9\left(-500\right)}}{2\times 9}$$

Multiply -4 times 9.

$$x=\frac{-\left(-155\right)±\sqrt{24025-36\left(-500\right)}}{2\times 9}$$

Multiply -36 times -500.

$$x=\frac{-\left(-155\right)±\sqrt{24025+18000}}{2\times 9}$$

Add 24025 to 18000.

$$x=\frac{-\left(-155\right)±\sqrt{42025}}{2\times 9}$$

Take the square root of 42025.

$$x=\frac{-\left(-155\right)±205}{2\times 9}$$

The opposite of -155 is 155.

$$x=\frac{155±205}{2\times 9}$$

Multiply 2 times 9.

$$x=\frac{155±205}{18}$$

Now solve the equation x=\frac{155±205}{18} when ± is plus. Add 155 to 205.

$$x=\frac{360}{18}$$

Divide 360 by 18.

$$x=20$$

Now solve the equation x=\frac{155±205}{18} when ± is minus. Subtract 205 from 155.

$$x=-\frac{50}{18}$$

Reduce the fraction \frac{-50}{18} to lowest terms by extracting and canceling out 2.

$$x=-\frac{25}{9}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 20 for x_{1} and -\frac{25}{9} for x_{2}.

$$9x^{2}-155x-500=9\left(x-20\right)\left(x-\left(-\frac{25}{9}\right)\right)$$

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

$$9x^{2}-155x-500=9\left(x-20\right)\left(x+\frac{25}{9}\right)$$

Add \frac{25}{9} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$9x^{2}-155x-500=9\left(x-20\right)\times \frac{9x+25}{9}$$

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

$$9x^{2}-155x-500=\left(x-20\right)\left(9x+25\right)$$

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

$$x ^ 2 -\frac{155}{9}x -\frac{500}{9} = 0$$

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

$$r + s = \frac{155}{9} rs = -\frac{500}{9}$$

Two numbers r and s sum up to \frac{155}{9} exactly when the average of the two numbers is \frac{1}{2}*\frac{155}{9} = \frac{155}{18}. 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>

$$r = \frac{155}{18} - u s = \frac{155}{18} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{500}{9}

$$(\frac{155}{18} - u) (\frac{155}{18} + u) = -\frac{500}{9}$$

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

$$\frac{24025}{324} - u^2 = -\frac{500}{9}$$

Simplify the expression by subtracting \frac{24025}{324} on both sides

$$-u^2 = -\frac{500}{9}-\frac{24025}{324} = -\frac{42025}{324}$$

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

$$u^2 = \frac{42025}{324} u = \pm\sqrt{\frac{42025}{324}} = \pm \frac{205}{18}$$

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

$$r =\frac{155}{18} - \frac{205}{18} = -2.778 s = \frac{155}{18} + \frac{205}{18} = 20$$

Evaluate

$$\left(x-20\right)\left(9x+25\right)$$