Solve 18x^2-x-4

$$18 x ^ { 2 } - x - 4$$

Factor

$$\left(2x-1\right)\left(9x+4\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 18x^{2}+ax+bx-4. To find a and b, set up a system to be solved.

$$a+b=-1 ab=18\left(-4\right)=-72$$

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 -72.

$$1,-72 2,-36 3,-24 4,-18 6,-12 8,-9$$

Calculate the sum for each pair.

$$1-72=-71 2-36=-34 3-24=-21 4-18=-14 6-12=-6 8-9=-1$$

The solution is the pair that gives sum -1.

$$a=-9 b=8$$

Rewrite 18x^{2}-x-4 as \left(18x^{2}-9x\right)+\left(8x-4\right).

$$\left(18x^{2}-9x\right)+\left(8x-4\right)$$

Factor out 9x in the first and 4 in the second group.

$$9x\left(2x-1\right)+4\left(2x-1\right)$$

Factor out common term 2x-1 by using distributive property.

$$\left(2x-1\right)\left(9x+4\right)$$

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.

$$18x^{2}-x-4=0$$

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.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\times 18\left(-4\right)}}{2\times 18}$$

Multiply -4 times 18.

$$x=\frac{-\left(-1\right)±\sqrt{1-72\left(-4\right)}}{2\times 18}$$

Multiply -72 times -4.

$$x=\frac{-\left(-1\right)±\sqrt{1+288}}{2\times 18}$$

Add 1 to 288.

$$x=\frac{-\left(-1\right)±\sqrt{289}}{2\times 18}$$

Take the square root of 289.

$$x=\frac{-\left(-1\right)±17}{2\times 18}$$

The opposite of -1 is 1.

$$x=\frac{1±17}{2\times 18}$$

Multiply 2 times 18.

$$x=\frac{1±17}{36}$$

Now solve the equation x=\frac{1±17}{36} when ± is plus. Add 1 to 17.

$$x=\frac{18}{36}$$

Reduce the fraction \frac{18}{36} to lowest terms by extracting and canceling out 18.

$$x=\frac{1}{2}$$

Now solve the equation x=\frac{1±17}{36} when ± is minus. Subtract 17 from 1.

$$x=-\frac{16}{36}$$

Reduce the fraction \frac{-16}{36} to lowest terms by extracting and canceling out 4.

$$x=-\frac{4}{9}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{1}{2} for x_{1} and -\frac{4}{9} for x_{2}.

$$18x^{2}-x-4=18\left(x-\frac{1}{2}\right)\left(x-\left(-\frac{4}{9}\right)\right)$$

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

$$18x^{2}-x-4=18\left(x-\frac{1}{2}\right)\left(x+\frac{4}{9}\right)$$

Subtract \frac{1}{2} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$18x^{2}-x-4=18\times \frac{2x-1}{2}\left(x+\frac{4}{9}\right)$$

Add \frac{4}{9} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$18x^{2}-x-4=18\times \frac{2x-1}{2}\times \frac{9x+4}{9}$$

Multiply \frac{2x-1}{2} times \frac{9x+4}{9} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$18x^{2}-x-4=18\times \frac{\left(2x-1\right)\left(9x+4\right)}{2\times 9}$$

Multiply 2 times 9.

$$18x^{2}-x-4=18\times \frac{\left(2x-1\right)\left(9x+4\right)}{18}$$

Cancel out 18, the greatest common factor in 18 and 18.

$$18x^{2}-x-4=\left(2x-1\right)\left(9x+4\right)$$

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 18

$$x ^ 2 -\frac{1}{18}x -\frac{2}{9} = 0$$

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

$$r + s = \frac{1}{18} rs = -\frac{2}{9}$$

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

$$r = \frac{1}{36} - u s = \frac{1}{36} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{2}{9}

$$(\frac{1}{36} - u) (\frac{1}{36} + u) = -\frac{2}{9}$$

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

$$\frac{1}{1296} - u^2 = -\frac{2}{9}$$

Simplify the expression by subtracting \frac{1}{1296} on both sides

$$-u^2 = -\frac{2}{9}-\frac{1}{1296} = -\frac{289}{1296}$$

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

$$u^2 = \frac{289}{1296} u = \pm\sqrt{\frac{289}{1296}} = \pm \frac{17}{36}$$

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

$$r =\frac{1}{36} - \frac{17}{36} = -0.444 s = \frac{1}{36} + \frac{17}{36} = 0.500$$

Evaluate

$$\left(2x-1\right)\left(9x+4\right)$$