Solve 7n^2-27n+18

$$7 n ^ { 2 } - 27 n + 18$$

Factor

$$\left(n-3\right)\left(7n-6\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 7n^{2}+an+bn+18. To find a and b, set up a system to be solved.

$$a+b=-27 ab=7\times 18=126$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 126.

$$-1,-126 -2,-63 -3,-42 -6,-21 -7,-18 -9,-14$$

Calculate the sum for each pair.

$$-1-126=-127 -2-63=-65 -3-42=-45 -6-21=-27 -7-18=-25 -9-14=-23$$

The solution is the pair that gives sum -27.

$$a=-21 b=-6$$

Rewrite 7n^{2}-27n+18 as \left(7n^{2}-21n\right)+\left(-6n+18\right).

$$\left(7n^{2}-21n\right)+\left(-6n+18\right)$$

Factor out 7n in the first and -6 in the second group.

$$7n\left(n-3\right)-6\left(n-3\right)$$

Factor out common term n-3 by using distributive property.

$$\left(n-3\right)\left(7n-6\right)$$

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.

$$7n^{2}-27n+18=0$$

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.

$$n=\frac{-\left(-27\right)±\sqrt{\left(-27\right)^{2}-4\times 7\times 18}}{2\times 7}$$

Square -27.

$$n=\frac{-\left(-27\right)±\sqrt{729-4\times 7\times 18}}{2\times 7}$$

Multiply -4 times 7.

$$n=\frac{-\left(-27\right)±\sqrt{729-28\times 18}}{2\times 7}$$

Multiply -28 times 18.

$$n=\frac{-\left(-27\right)±\sqrt{729-504}}{2\times 7}$$

Add 729 to -504.

$$n=\frac{-\left(-27\right)±\sqrt{225}}{2\times 7}$$

Take the square root of 225.

$$n=\frac{-\left(-27\right)±15}{2\times 7}$$

The opposite of -27 is 27.

$$n=\frac{27±15}{2\times 7}$$

Multiply 2 times 7.

$$n=\frac{27±15}{14}$$

Now solve the equation n=\frac{27±15}{14} when ± is plus. Add 27 to 15.

$$n=\frac{42}{14}$$

Divide 42 by 14.

$$n=3$$

Now solve the equation n=\frac{27±15}{14} when ± is minus. Subtract 15 from 27.

$$n=\frac{12}{14}$$

Reduce the fraction \frac{12}{14} to lowest terms by extracting and canceling out 2.

$$n=\frac{6}{7}$$

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

$$7n^{2}-27n+18=7\left(n-3\right)\left(n-\frac{6}{7}\right)$$

Subtract \frac{6}{7} from n by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$7n^{2}-27n+18=7\left(n-3\right)\times \frac{7n-6}{7}$$

Cancel out 7, the greatest common factor in 7 and 7.

$$7n^{2}-27n+18=\left(n-3\right)\left(7n-6\right)$$

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 7

$$x ^ 2 -\frac{27}{7}x +\frac{18}{7} = 0$$

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

$$r + s = \frac{27}{7} rs = \frac{18}{7}$$

Two numbers r and s sum up to \frac{27}{7} exactly when the average of the two numbers is \frac{1}{2}*\frac{27}{7} = \frac{27}{14}. 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>

$$r = \frac{27}{14} - u s = \frac{27}{14} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{18}{7}

$$(\frac{27}{14} - u) (\frac{27}{14} + u) = \frac{18}{7}$$

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

$$\frac{729}{196} - u^2 = \frac{18}{7}$$

Simplify the expression by subtracting \frac{729}{196} on both sides

$$-u^2 = \frac{18}{7}-\frac{729}{196} = -\frac{225}{196}$$

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

$$u^2 = \frac{225}{196} u = \pm\sqrt{\frac{225}{196}} = \pm \frac{15}{14}$$

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

$$r =\frac{27}{14} - \frac{15}{14} = 0.857 s = \frac{27}{14} + \frac{15}{14} = 3$$

Evaluate

$$\left(n-3\right)\left(7n-6\right)$$