Solve 18x^2+59x-14

$$18 x ^ { 2 } + 59 x - 14$$

Factor

$$\left(9x-2\right)\left(2x+7\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 18x^{2}+ax+bx-14. To find a and b, set up a system to be solved.

$$a+b=59 ab=18\left(-14\right)=-252$$

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 -252.

$$-1,252 -2,126 -3,84 -4,63 -6,42 -7,36 -9,28 -12,21 -14,18$$

Calculate the sum for each pair.

$$-1+252=251 -2+126=124 -3+84=81 -4+63=59 -6+42=36 -7+36=29 -9+28=19 -12+21=9 -14+18=4$$

The solution is the pair that gives sum 59.

$$a=-4 b=63$$

Rewrite 18x^{2}+59x-14 as \left(18x^{2}-4x\right)+\left(63x-14\right).

$$\left(18x^{2}-4x\right)+\left(63x-14\right)$$

Factor out 2x in the first and 7 in the second group.

$$2x\left(9x-2\right)+7\left(9x-2\right)$$

Factor out common term 9x-2 by using distributive property.

$$\left(9x-2\right)\left(2x+7\right)$$

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.

$$18x^{2}+59x-14=0$$

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.

$$x=\frac{-59±\sqrt{59^{2}-4\times 18\left(-14\right)}}{2\times 18}$$

Square 59.

$$x=\frac{-59±\sqrt{3481-4\times 18\left(-14\right)}}{2\times 18}$$

Multiply -4 times 18.

$$x=\frac{-59±\sqrt{3481-72\left(-14\right)}}{2\times 18}$$

Multiply -72 times -14.

$$x=\frac{-59±\sqrt{3481+1008}}{2\times 18}$$

Add 3481 to 1008.

$$x=\frac{-59±\sqrt{4489}}{2\times 18}$$

Take the square root of 4489.

$$x=\frac{-59±67}{2\times 18}$$

Multiply 2 times 18.

$$x=\frac{-59±67}{36}$$

Now solve the equation x=\frac{-59±67}{36} when ± is plus. Add -59 to 67.

$$x=\frac{8}{36}$$

Reduce the fraction \frac{8}{36} to lowest terms by extracting and canceling out 4.

$$x=\frac{2}{9}$$

Now solve the equation x=\frac{-59±67}{36} when ± is minus. Subtract 67 from -59.

$$x=-\frac{126}{36}$$

Reduce the fraction \frac{-126}{36} to lowest terms by extracting and canceling out 18.

$$x=-\frac{7}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{2}{9} for x_{1} and -\frac{7}{2} for x_{2}.

$$18x^{2}+59x-14=18\left(x-\frac{2}{9}\right)\left(x-\left(-\frac{7}{2}\right)\right)$$

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

$$18x^{2}+59x-14=18\left(x-\frac{2}{9}\right)\left(x+\frac{7}{2}\right)$$

Subtract \frac{2}{9} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$18x^{2}+59x-14=18\times \frac{9x-2}{9}\left(x+\frac{7}{2}\right)$$

Add \frac{7}{2} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$18x^{2}+59x-14=18\times \frac{9x-2}{9}\times \frac{2x+7}{2}$$

Multiply \frac{9x-2}{9} times \frac{2x+7}{2} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$18x^{2}+59x-14=18\times \frac{\left(9x-2\right)\left(2x+7\right)}{9\times 2}$$

Multiply 9 times 2.

$$18x^{2}+59x-14=18\times \frac{\left(9x-2\right)\left(2x+7\right)}{18}$$

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

$$18x^{2}+59x-14=\left(9x-2\right)\left(2x+7\right)$$

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

$$x ^ 2 +\frac{59}{18}x -\frac{7}{9} = 0$$

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

$$r + s = -\frac{59}{18} rs = -\frac{7}{9}$$

Two numbers r and s sum up to -\frac{59}{18} exactly when the average of the two numbers is \frac{1}{2}*-\frac{59}{18} = -\frac{59}{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>

$$r = -\frac{59}{36} - u s = -\frac{59}{36} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{7}{9}

$$(-\frac{59}{36} - u) (-\frac{59}{36} + u) = -\frac{7}{9}$$

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

$$\frac{3481}{1296} - u^2 = -\frac{7}{9}$$

Simplify the expression by subtracting \frac{3481}{1296} on both sides

$$-u^2 = -\frac{7}{9}-\frac{3481}{1296} = -\frac{4489}{1296}$$

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

$$u^2 = \frac{4489}{1296} u = \pm\sqrt{\frac{4489}{1296}} = \pm \frac{67}{36}$$

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

$$r =-\frac{59}{36} - \frac{67}{36} = -3.500 s = -\frac{59}{36} + \frac{67}{36} = 0.222$$

Evaluate

$$\left(9x-2\right)\left(2x+7\right)$$