Solve 6n^2-25n+21

$$6 n ^ { 2 } - 25 n + 21$$

Factor

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

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 6n^{2}+an+bn+21. To find a and b, set up a system to be solved.

$$a+b=-25 ab=6\times 21=126$$

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 -25.

$$a=-18 b=-7$$

Rewrite 6n^{2}-25n+21 as \left(6n^{2}-18n\right)+\left(-7n+21\right).

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

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

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

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

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

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.

$$6n^{2}-25n+21=0$$

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.

$$n=\frac{-\left(-25\right)±\sqrt{\left(-25\right)^{2}-4\times 6\times 21}}{2\times 6}$$

Square -25.

$$n=\frac{-\left(-25\right)±\sqrt{625-4\times 6\times 21}}{2\times 6}$$

Multiply -4 times 6.

$$n=\frac{-\left(-25\right)±\sqrt{625-24\times 21}}{2\times 6}$$

Multiply -24 times 21.

$$n=\frac{-\left(-25\right)±\sqrt{625-504}}{2\times 6}$$

Add 625 to -504.

$$n=\frac{-\left(-25\right)±\sqrt{121}}{2\times 6}$$

Take the square root of 121.

$$n=\frac{-\left(-25\right)±11}{2\times 6}$$

The opposite of -25 is 25.

$$n=\frac{25±11}{2\times 6}$$

Multiply 2 times 6.

$$n=\frac{25±11}{12}$$

Now solve the equation n=\frac{25±11}{12} when ± is plus. Add 25 to 11.

$$n=\frac{36}{12}$$

Divide 36 by 12.

$$n=3$$

Now solve the equation n=\frac{25±11}{12} when ± is minus. Subtract 11 from 25.

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

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

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

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{7}{6} for x_{2}.

$$6n^{2}-25n+21=6\left(n-3\right)\left(n-\frac{7}{6}\right)$$

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

$$6n^{2}-25n+21=6\left(n-3\right)\times \frac{6n-7}{6}$$

Cancel out 6, the greatest common factor in 6 and 6.

$$6n^{2}-25n+21=\left(n-3\right)\left(6n-7\right)$$

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 6

$$x ^ 2 -\frac{25}{6}x +\frac{7}{2} = 0$$

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

$$r + s = \frac{25}{6} rs = \frac{7}{2}$$

Two numbers r and s sum up to \frac{25}{6} exactly when the average of the two numbers is \frac{1}{2}*\frac{25}{6} = \frac{25}{12}. 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>

$$r = \frac{25}{12} - u s = \frac{25}{12} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{7}{2}

$$(\frac{25}{12} - u) (\frac{25}{12} + u) = \frac{7}{2}$$

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

$$\frac{625}{144} - u^2 = \frac{7}{2}$$

Simplify the expression by subtracting \frac{625}{144} on both sides

$$-u^2 = \frac{7}{2}-\frac{625}{144} = -\frac{121}{144}$$

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

$$u^2 = \frac{121}{144} u = \pm\sqrt{\frac{121}{144}} = \pm \frac{11}{12}$$

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

$$r =\frac{25}{12} - \frac{11}{12} = 1.167 s = \frac{25}{12} + \frac{11}{12} = 3.000$$

Evaluate

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