Solve 54k^2+54k-45

$$54 k ^ { 2 } + 54 k - 45$$

Factor

$$54\left(k-\left(-\frac{\sqrt{39}}{6}-\frac{1}{2}\right)\right)\left(k-\left(\frac{\sqrt{39}}{6}-\frac{1}{2}\right)\right)$$

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.

$$54k^{2}+54k-45=0$$

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.

$$k=\frac{-54±\sqrt{54^{2}-4\times 54\left(-45\right)}}{2\times 54}$$

Square 54.

$$k=\frac{-54±\sqrt{2916-4\times 54\left(-45\right)}}{2\times 54}$$

Multiply -4 times 54.

$$k=\frac{-54±\sqrt{2916-216\left(-45\right)}}{2\times 54}$$

Multiply -216 times -45.

$$k=\frac{-54±\sqrt{2916+9720}}{2\times 54}$$

Add 2916 to 9720.

$$k=\frac{-54±\sqrt{12636}}{2\times 54}$$

Take the square root of 12636.

$$k=\frac{-54±18\sqrt{39}}{2\times 54}$$

Multiply 2 times 54.

$$k=\frac{-54±18\sqrt{39}}{108}$$

Now solve the equation k=\frac{-54±18\sqrt{39}}{108} when ± is plus. Add -54 to 18\sqrt{39}.

$$k=\frac{18\sqrt{39}-54}{108}$$

Divide -54+18\sqrt{39} by 108.

$$k=\frac{\sqrt{39}}{6}-\frac{1}{2}$$

Now solve the equation k=\frac{-54±18\sqrt{39}}{108} when ± is minus. Subtract 18\sqrt{39} from -54.

$$k=\frac{-18\sqrt{39}-54}{108}$$

Divide -54-18\sqrt{39} by 108.

$$k=-\frac{\sqrt{39}}{6}-\frac{1}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{1}{2}+\frac{\sqrt{39}}{6} for x_{1} and -\frac{1}{2}-\frac{\sqrt{39}}{6} for x_{2}.

$$54k^{2}+54k-45=54\left(k-\left(\frac{\sqrt{39}}{6}-\frac{1}{2}\right)\right)\left(k-\left(-\frac{\sqrt{39}}{6}-\frac{1}{2}\right)\right)$$

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 54

$$x ^ 2 +1x -\frac{5}{6} = 0$$

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

$$r + s = -1 rs = -\frac{5}{6}$$

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

$$r = -\frac{1}{2} - u s = -\frac{1}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{5}{6}

$$(-\frac{1}{2} - u) (-\frac{1}{2} + u) = -\frac{5}{6}$$

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

$$\frac{1}{4} - u^2 = -\frac{5}{6}$$

Simplify the expression by subtracting \frac{1}{4} on both sides

$$-u^2 = -\frac{5}{6}-\frac{1}{4} = -\frac{13}{12}$$

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

$$u^2 = \frac{13}{12} u = \pm\sqrt{\frac{13}{12}} = \pm \frac{\sqrt{13}}{\sqrt{12}}$$

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

$$r =-\frac{1}{2} - \frac{\sqrt{13}}{\sqrt{12}} = -1.541 s = -\frac{1}{2} + \frac{\sqrt{13}}{\sqrt{12}} = 0.541$$

Evaluate

$$54k^{2}+54k-45$$