Solve 8y^2+30y-27

$$8 y ^ { 2 } + 30 y - 27$$

Factor

$$\left(4y-3\right)\left(2y+9\right)$$

Solution Steps

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

$$a+b=30 ab=8\left(-27\right)=-216$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -216.

$$-1,216 -2,108 -3,72 -4,54 -6,36 -8,27 -9,24 -12,18$$

Calculate the sum for each pair.

$$-1+216=215 -2+108=106 -3+72=69 -4+54=50 -6+36=30 -8+27=19 -9+24=15 -12+18=6$$

The solution is the pair that gives sum 30.

$$a=-6 b=36$$

Rewrite 8y^{2}+30y-27 as \left(8y^{2}-6y\right)+\left(36y-27\right).

$$\left(8y^{2}-6y\right)+\left(36y-27\right)$$

Factor out 2y in the first and 9 in the second group.

$$2y\left(4y-3\right)+9\left(4y-3\right)$$

Factor out common term 4y-3 by using distributive property.

$$\left(4y-3\right)\left(2y+9\right)$$

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.

$$8y^{2}+30y-27=0$$

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.

$$y=\frac{-30±\sqrt{30^{2}-4\times 8\left(-27\right)}}{2\times 8}$$

Square 30.

$$y=\frac{-30±\sqrt{900-4\times 8\left(-27\right)}}{2\times 8}$$

Multiply -4 times 8.

$$y=\frac{-30±\sqrt{900-32\left(-27\right)}}{2\times 8}$$

Multiply -32 times -27.

$$y=\frac{-30±\sqrt{900+864}}{2\times 8}$$

Add 900 to 864.

$$y=\frac{-30±\sqrt{1764}}{2\times 8}$$

Take the square root of 1764.

$$y=\frac{-30±42}{2\times 8}$$

Multiply 2 times 8.

$$y=\frac{-30±42}{16}$$

Now solve the equation y=\frac{-30±42}{16} when ± is plus. Add -30 to 42.

$$y=\frac{12}{16}$$

Reduce the fraction \frac{12}{16} to lowest terms by extracting and canceling out 4.

$$y=\frac{3}{4}$$

Now solve the equation y=\frac{-30±42}{16} when ± is minus. Subtract 42 from -30.

$$y=-\frac{72}{16}$$

Reduce the fraction \frac{-72}{16} to lowest terms by extracting and canceling out 8.

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

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{3}{4} for x_{1} and -\frac{9}{2} for x_{2}.

$$8y^{2}+30y-27=8\left(y-\frac{3}{4}\right)\left(y-\left(-\frac{9}{2}\right)\right)$$

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

$$8y^{2}+30y-27=8\left(y-\frac{3}{4}\right)\left(y+\frac{9}{2}\right)$$

Subtract \frac{3}{4} from y by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$8y^{2}+30y-27=8\times \frac{4y-3}{4}\left(y+\frac{9}{2}\right)$$

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

$$8y^{2}+30y-27=8\times \frac{4y-3}{4}\times \frac{2y+9}{2}$$

Multiply \frac{4y-3}{4} times \frac{2y+9}{2} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$8y^{2}+30y-27=8\times \frac{\left(4y-3\right)\left(2y+9\right)}{4\times 2}$$

Multiply 4 times 2.

$$8y^{2}+30y-27=8\times \frac{\left(4y-3\right)\left(2y+9\right)}{8}$$

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

$$8y^{2}+30y-27=\left(4y-3\right)\left(2y+9\right)$$

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 8

$$x ^ 2 +\frac{15}{4}x -\frac{27}{8} = 0$$

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

$$r + s = -\frac{15}{4} rs = -\frac{27}{8}$$

Two numbers r and s sum up to -\frac{15}{4} exactly when the average of the two numbers is \frac{1}{2}*-\frac{15}{4} = -\frac{15}{8}. 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>

$$r = -\frac{15}{8} - u s = -\frac{15}{8} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{27}{8}

$$(-\frac{15}{8} - u) (-\frac{15}{8} + u) = -\frac{27}{8}$$

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

$$\frac{225}{64} - u^2 = -\frac{27}{8}$$

Simplify the expression by subtracting \frac{225}{64} on both sides

$$-u^2 = -\frac{27}{8}-\frac{225}{64} = -\frac{441}{64}$$

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

$$u^2 = \frac{441}{64} u = \pm\sqrt{\frac{441}{64}} = \pm \frac{21}{8}$$

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

$$r =-\frac{15}{8} - \frac{21}{8} = -4.500 s = -\frac{15}{8} + \frac{21}{8} = 0.750$$

Evaluate

$$\left(4y-3\right)\left(2y+9\right)$$