Solve 4y^2-24y+27

$$4 y ^ { 2 } - 24 y + 27$$

Factor

$$\left(2y-9\right)\left(2y-3\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 4y^{2}+ay+by+27. To find a and b, set up a system to be solved.

$$a+b=-24 ab=4\times 27=108$$

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 108.

$$-1,-108 -2,-54 -3,-36 -4,-27 -6,-18 -9,-12$$

Calculate the sum for each pair.

$$-1-108=-109 -2-54=-56 -3-36=-39 -4-27=-31 -6-18=-24 -9-12=-21$$

The solution is the pair that gives sum -24.

$$a=-18 b=-6$$

Rewrite 4y^{2}-24y+27 as \left(4y^{2}-18y\right)+\left(-6y+27\right).

$$\left(4y^{2}-18y\right)+\left(-6y+27\right)$$

Factor out 2y in the first and -3 in the second group.

$$2y\left(2y-9\right)-3\left(2y-9\right)$$

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

$$\left(2y-9\right)\left(2y-3\right)$$

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.

$$4y^{2}-24y+27=0$$

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.

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

Square -24.

$$y=\frac{-\left(-24\right)±\sqrt{576-4\times 4\times 27}}{2\times 4}$$

Multiply -4 times 4.

$$y=\frac{-\left(-24\right)±\sqrt{576-16\times 27}}{2\times 4}$$

Multiply -16 times 27.

$$y=\frac{-\left(-24\right)±\sqrt{576-432}}{2\times 4}$$

Add 576 to -432.

$$y=\frac{-\left(-24\right)±\sqrt{144}}{2\times 4}$$

Take the square root of 144.

$$y=\frac{-\left(-24\right)±12}{2\times 4}$$

The opposite of -24 is 24.

$$y=\frac{24±12}{2\times 4}$$

Multiply 2 times 4.

$$y=\frac{24±12}{8}$$

Now solve the equation y=\frac{24±12}{8} when ± is plus. Add 24 to 12.

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

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

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

Now solve the equation y=\frac{24±12}{8} when ± is minus. Subtract 12 from 24.

$$y=\frac{12}{8}$$

Reduce the fraction \frac{12}{8} to lowest terms by extracting and canceling out 4.

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

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{9}{2} for x_{1} and \frac{3}{2} for x_{2}.

$$4y^{2}-24y+27=4\left(y-\frac{9}{2}\right)\left(y-\frac{3}{2}\right)$$

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

$$4y^{2}-24y+27=4\times \frac{2y-9}{2}\left(y-\frac{3}{2}\right)$$

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

$$4y^{2}-24y+27=4\times \frac{2y-9}{2}\times \frac{2y-3}{2}$$

Multiply \frac{2y-9}{2} times \frac{2y-3}{2} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$4y^{2}-24y+27=4\times \frac{\left(2y-9\right)\left(2y-3\right)}{2\times 2}$$

Multiply 2 times 2.

$$4y^{2}-24y+27=4\times \frac{\left(2y-9\right)\left(2y-3\right)}{4}$$

Cancel out 4, the greatest common factor in 4 and 4.

$$4y^{2}-24y+27=\left(2y-9\right)\left(2y-3\right)$$

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 4

$$x ^ 2 -6x +\frac{27}{4} = 0$$

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

$$r + s = 6 rs = \frac{27}{4}$$

Two numbers r and s sum up to 6 exactly when the average of the two numbers is \frac{1}{2}*6 = 3. 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>

$$r = 3 - u s = 3 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{27}{4}

$$(3 - u) (3 + u) = \frac{27}{4}$$

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

$$9 - u^2 = \frac{27}{4}$$

Simplify the expression by subtracting 9 on both sides

$$-u^2 = \frac{27}{4}-9 = -\frac{9}{4}$$

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

$$u^2 = \frac{9}{4} u = \pm\sqrt{\frac{9}{4}} = \pm \frac{3}{2}$$

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

$$r =3 - \frac{3}{2} = 1.500 s = 3 + \frac{3}{2} = 4.500$$

Evaluate

$$\left(2y-9\right)\left(2y-3\right)$$