Solve 54x^2+78x-81

$$54 { x }^{ 2 } +78x-81$$

Factor

$$54\left(x-\frac{-\sqrt{655}-13}{18}\right)\left(x-\frac{\sqrt{655}-13}{18}\right)$$

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.

$$54x^{2}+78x-81=0$$

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.

$$x=\frac{-78±\sqrt{78^{2}-4\times 54\left(-81\right)}}{2\times 54}$$

Square 78.

$$x=\frac{-78±\sqrt{6084-4\times 54\left(-81\right)}}{2\times 54}$$

Multiply -4 times 54.

$$x=\frac{-78±\sqrt{6084-216\left(-81\right)}}{2\times 54}$$

Multiply -216 times -81.

$$x=\frac{-78±\sqrt{6084+17496}}{2\times 54}$$

Add 6084 to 17496.

$$x=\frac{-78±\sqrt{23580}}{2\times 54}$$

Take the square root of 23580.

$$x=\frac{-78±6\sqrt{655}}{2\times 54}$$

Multiply 2 times 54.

$$x=\frac{-78±6\sqrt{655}}{108}$$

Now solve the equation x=\frac{-78±6\sqrt{655}}{108} when ± is plus. Add -78 to 6\sqrt{655}.

$$x=\frac{6\sqrt{655}-78}{108}$$

Divide -78+6\sqrt{655} by 108.

$$x=\frac{\sqrt{655}-13}{18}$$

Now solve the equation x=\frac{-78±6\sqrt{655}}{108} when ± is minus. Subtract 6\sqrt{655} from -78.

$$x=\frac{-6\sqrt{655}-78}{108}$$

Divide -78-6\sqrt{655} by 108.

$$x=\frac{-\sqrt{655}-13}{18}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-13+\sqrt{655}}{18} for x_{1} and \frac{-13-\sqrt{655}}{18} for x_{2}.

$$54x^{2}+78x-81=54\left(x-\frac{\sqrt{655}-13}{18}\right)\left(x-\frac{-\sqrt{655}-13}{18}\right)$$

Evaluate

$$54x^{2}+78x-81$$