Solve 3y^2-27y-45

$$3 y ^ { 2 } - 27 y - 45$$

Factor

$$3\left(y-\frac{9-\sqrt{141}}{2}\right)\left(y-\frac{\sqrt{141}+9}{2}\right)$$

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.

$$3y^{2}-27y-45=0$$

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.

$$y=\frac{-\left(-27\right)±\sqrt{\left(-27\right)^{2}-4\times 3\left(-45\right)}}{2\times 3}$$

Square -27.

$$y=\frac{-\left(-27\right)±\sqrt{729-4\times 3\left(-45\right)}}{2\times 3}$$

Multiply -4 times 3.

$$y=\frac{-\left(-27\right)±\sqrt{729-12\left(-45\right)}}{2\times 3}$$

Multiply -12 times -45.

$$y=\frac{-\left(-27\right)±\sqrt{729+540}}{2\times 3}$$

Add 729 to 540.

$$y=\frac{-\left(-27\right)±\sqrt{1269}}{2\times 3}$$

Take the square root of 1269.

$$y=\frac{-\left(-27\right)±3\sqrt{141}}{2\times 3}$$

The opposite of -27 is 27.

$$y=\frac{27±3\sqrt{141}}{2\times 3}$$

Multiply 2 times 3.

$$y=\frac{27±3\sqrt{141}}{6}$$

Now solve the equation y=\frac{27±3\sqrt{141}}{6} when ± is plus. Add 27 to 3\sqrt{141}.

$$y=\frac{3\sqrt{141}+27}{6}$$

Divide 27+3\sqrt{141} by 6.

$$y=\frac{\sqrt{141}+9}{2}$$

Now solve the equation y=\frac{27±3\sqrt{141}}{6} when ± is minus. Subtract 3\sqrt{141} from 27.

$$y=\frac{27-3\sqrt{141}}{6}$$

Divide 27-3\sqrt{141} by 6.

$$y=\frac{9-\sqrt{141}}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{9+\sqrt{141}}{2} for x_{1} and \frac{9-\sqrt{141}}{2} for x_{2}.

$$3y^{2}-27y-45=3\left(y-\frac{\sqrt{141}+9}{2}\right)\left(y-\frac{9-\sqrt{141}}{2}\right)$$

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 3

$$x ^ 2 -9x -15 = 0$$

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

$$r + s = 9 rs = -15$$

Two numbers r and s sum up to 9 exactly when the average of the two numbers is \frac{1}{2}*9 = \frac{9}{2}. 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>

$$r = \frac{9}{2} - u s = \frac{9}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -15

$$(\frac{9}{2} - u) (\frac{9}{2} + u) = -15$$

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

$$\frac{81}{4} - u^2 = -15$$

Simplify the expression by subtracting \frac{81}{4} on both sides

$$-u^2 = -15-\frac{81}{4} = -\frac{141}{4}$$

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

$$u^2 = \frac{141}{4} u = \pm\sqrt{\frac{141}{4}} = \pm \frac{\sqrt{141}}{2}$$

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

$$r =\frac{9}{2} - \frac{\sqrt{141}}{2} = -1.437 s = \frac{9}{2} + \frac{\sqrt{141}}{2} = 10.437$$

Evaluate

$$3\left(y^{2}-9y-15\right)$$