Solve 14k^2-9k-18

$$14 k ^ { 2 } - 9 k - 18$$

Factor

$$\left(2k-3\right)\left(7k+6\right)$$

Solution Steps

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

$$a+b=-9 ab=14\left(-18\right)=-252$$

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 -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 -9.

$$a=-21 b=12$$

Rewrite 14k^{2}-9k-18 as \left(14k^{2}-21k\right)+\left(12k-18\right).

$$\left(14k^{2}-21k\right)+\left(12k-18\right)$$

Factor out 7k in the first and 6 in the second group.

$$7k\left(2k-3\right)+6\left(2k-3\right)$$

Factor out common term 2k-3 by using distributive property.

$$\left(2k-3\right)\left(7k+6\right)$$

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.

$$14k^{2}-9k-18=0$$

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.

$$k=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\times 14\left(-18\right)}}{2\times 14}$$

Square -9.

$$k=\frac{-\left(-9\right)±\sqrt{81-4\times 14\left(-18\right)}}{2\times 14}$$

Multiply -4 times 14.

$$k=\frac{-\left(-9\right)±\sqrt{81-56\left(-18\right)}}{2\times 14}$$

Multiply -56 times -18.

$$k=\frac{-\left(-9\right)±\sqrt{81+1008}}{2\times 14}$$

Add 81 to 1008.

$$k=\frac{-\left(-9\right)±\sqrt{1089}}{2\times 14}$$

Take the square root of 1089.

$$k=\frac{-\left(-9\right)±33}{2\times 14}$$

The opposite of -9 is 9.

$$k=\frac{9±33}{2\times 14}$$

Multiply 2 times 14.

$$k=\frac{9±33}{28}$$

Now solve the equation k=\frac{9±33}{28} when ± is plus. Add 9 to 33.

$$k=\frac{42}{28}$$

Reduce the fraction \frac{42}{28} to lowest terms by extracting and canceling out 14.

$$k=\frac{3}{2}$$

Now solve the equation k=\frac{9±33}{28} when ± is minus. Subtract 33 from 9.

$$k=-\frac{24}{28}$$

Reduce the fraction \frac{-24}{28} to lowest terms by extracting and canceling out 4.

$$k=-\frac{6}{7}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{3}{2} for x_{1} and -\frac{6}{7} for x_{2}.

$$14k^{2}-9k-18=14\left(k-\frac{3}{2}\right)\left(k-\left(-\frac{6}{7}\right)\right)$$

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

$$14k^{2}-9k-18=14\left(k-\frac{3}{2}\right)\left(k+\frac{6}{7}\right)$$

Subtract \frac{3}{2} from k by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$14k^{2}-9k-18=14\times \frac{2k-3}{2}\left(k+\frac{6}{7}\right)$$

Add \frac{6}{7} to k by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$14k^{2}-9k-18=14\times \frac{2k-3}{2}\times \frac{7k+6}{7}$$

Multiply \frac{2k-3}{2} times \frac{7k+6}{7} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$14k^{2}-9k-18=14\times \frac{\left(2k-3\right)\left(7k+6\right)}{2\times 7}$$

Multiply 2 times 7.

$$14k^{2}-9k-18=14\times \frac{\left(2k-3\right)\left(7k+6\right)}{14}$$

Cancel out 14, the greatest common factor in 14 and 14.

$$14k^{2}-9k-18=\left(2k-3\right)\left(7k+6\right)$$

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 14

$$x ^ 2 -\frac{9}{14}x -\frac{9}{7} = 0$$

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

$$r + s = \frac{9}{14} rs = -\frac{9}{7}$$

Two numbers r and s sum up to \frac{9}{14} exactly when the average of the two numbers is \frac{1}{2}*\frac{9}{14} = \frac{9}{28}. 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>

$$r = \frac{9}{28} - u s = \frac{9}{28} + u$$

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

$$(\frac{9}{28} - u) (\frac{9}{28} + u) = -\frac{9}{7}$$

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

$$\frac{81}{784} - u^2 = -\frac{9}{7}$$

Simplify the expression by subtracting \frac{81}{784} on both sides

$$-u^2 = -\frac{9}{7}-\frac{81}{784} = -\frac{1089}{784}$$

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

$$u^2 = \frac{1089}{784} u = \pm\sqrt{\frac{1089}{784}} = \pm \frac{33}{28}$$

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

$$r =\frac{9}{28} - \frac{33}{28} = -0.857 s = \frac{9}{28} + \frac{33}{28} = 1.500$$

Evaluate

$$\left(2k-3\right)\left(7k+6\right)$$