Solve 24x^2-81x+27

$$24 x ^ { 2 } - 81 x + 27$$

Factor

$$3\left(x-3\right)\left(8x-3\right)$$

Solution Steps

Factor out 3.

$$3\left(8x^{2}-27x+9\right)$$

Consider 8x^{2}-27x+9. Factor the expression by grouping. First, the expression needs to be rewritten as 8x^{2}+ax+bx+9. To find a and b, set up a system to be solved.

$$a+b=-27 ab=8\times 9=72$$

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 72.

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

Calculate the sum for each pair.

$$-1-72=-73 -2-36=-38 -3-24=-27 -4-18=-22 -6-12=-18 -8-9=-17$$

The solution is the pair that gives sum -27.

$$a=-24 b=-3$$

Rewrite 8x^{2}-27x+9 as \left(8x^{2}-24x\right)+\left(-3x+9\right).

$$\left(8x^{2}-24x\right)+\left(-3x+9\right)$$

Factor out 8x in the first and -3 in the second group.

$$8x\left(x-3\right)-3\left(x-3\right)$$

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

$$\left(x-3\right)\left(8x-3\right)$$

Rewrite the complete factored expression.

$$3\left(x-3\right)\left(8x-3\right)$$

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.

$$24x^{2}-81x+27=0$$

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.

$$x=\frac{-\left(-81\right)±\sqrt{\left(-81\right)^{2}-4\times 24\times 27}}{2\times 24}$$

Square -81.

$$x=\frac{-\left(-81\right)±\sqrt{6561-4\times 24\times 27}}{2\times 24}$$

Multiply -4 times 24.

$$x=\frac{-\left(-81\right)±\sqrt{6561-96\times 27}}{2\times 24}$$

Multiply -96 times 27.

$$x=\frac{-\left(-81\right)±\sqrt{6561-2592}}{2\times 24}$$

Add 6561 to -2592.

$$x=\frac{-\left(-81\right)±\sqrt{3969}}{2\times 24}$$

Take the square root of 3969.

$$x=\frac{-\left(-81\right)±63}{2\times 24}$$

The opposite of -81 is 81.

$$x=\frac{81±63}{2\times 24}$$

Multiply 2 times 24.

$$x=\frac{81±63}{48}$$

Now solve the equation x=\frac{81±63}{48} when ± is plus. Add 81 to 63.

$$x=\frac{144}{48}$$

Divide 144 by 48.

$$x=3$$

Now solve the equation x=\frac{81±63}{48} when ± is minus. Subtract 63 from 81.

$$x=\frac{18}{48}$$

Reduce the fraction \frac{18}{48} to lowest terms by extracting and canceling out 6.

$$x=\frac{3}{8}$$

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{3}{8} for x_{2}.

$$24x^{2}-81x+27=24\left(x-3\right)\left(x-\frac{3}{8}\right)$$

Subtract \frac{3}{8} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$24x^{2}-81x+27=24\left(x-3\right)\times \frac{8x-3}{8}$$

Cancel out 8, the greatest common factor in 24 and 8.

$$24x^{2}-81x+27=3\left(x-3\right)\left(8x-3\right)$$

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 24

$$x ^ 2 -\frac{27}{8}x +\frac{9}{8} = 0$$

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

$$r + s = \frac{27}{8} rs = \frac{9}{8}$$

Two numbers r and s sum up to \frac{27}{8} exactly when the average of the two numbers is \frac{1}{2}*\frac{27}{8} = \frac{27}{16}. 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>

$$r = \frac{27}{16} - u s = \frac{27}{16} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{9}{8}

$$(\frac{27}{16} - u) (\frac{27}{16} + u) = \frac{9}{8}$$

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

$$\frac{729}{256} - u^2 = \frac{9}{8}$$

Simplify the expression by subtracting \frac{729}{256} on both sides

$$-u^2 = \frac{9}{8}-\frac{729}{256} = -\frac{441}{256}$$

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

$$u^2 = \frac{441}{256} u = \pm\sqrt{\frac{441}{256}} = \pm \frac{21}{16}$$

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

$$r =\frac{27}{16} - \frac{21}{16} = 0.375 s = \frac{27}{16} + \frac{21}{16} = 3$$

Evaluate

$$3\left(x-3\right)\left(8x-3\right)$$