Solve 3x^2-54x=-204

$$3 x ^ { 2 } - 54 x = - 204$$

Solve for x

$$x=\sqrt{13}+9\approx 12.605551275$$

$$x=9-\sqrt{13}\approx 5.394448725$$

Steps Using the Quadratic Formula

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.

$$3x^{2}-54x=-204$$

Add 204 to both sides of the equation.

$$3x^{2}-54x-\left(-204\right)=-204-\left(-204\right)$$

Subtracting -204 from itself leaves 0.

$$3x^{2}-54x-\left(-204\right)=0$$

Subtract -204 from 0.

$$3x^{2}-54x+204=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -54 for b, and 204 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-54\right)±\sqrt{\left(-54\right)^{2}-4\times 3\times 204}}{2\times 3}$$

Square -54.

$$x=\frac{-\left(-54\right)±\sqrt{2916-4\times 3\times 204}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-\left(-54\right)±\sqrt{2916-12\times 204}}{2\times 3}$$

Multiply -12 times 204.

$$x=\frac{-\left(-54\right)±\sqrt{2916-2448}}{2\times 3}$$

Add 2916 to -2448.

$$x=\frac{-\left(-54\right)±\sqrt{468}}{2\times 3}$$

Take the square root of 468.

$$x=\frac{-\left(-54\right)±6\sqrt{13}}{2\times 3}$$

The opposite of -54 is 54.

$$x=\frac{54±6\sqrt{13}}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{54±6\sqrt{13}}{6}$$

Now solve the equation x=\frac{54±6\sqrt{13}}{6} when ± is plus. Add 54 to 6\sqrt{13}.

$$x=\frac{6\sqrt{13}+54}{6}$$

Divide 54+6\sqrt{13} by 6.

$$x=\sqrt{13}+9$$

Now solve the equation x=\frac{54±6\sqrt{13}}{6} when ± is minus. Subtract 6\sqrt{13} from 54.

$$x=\frac{54-6\sqrt{13}}{6}$$

Divide 54-6\sqrt{13} by 6.

$$x=9-\sqrt{13}$$

The equation is now solved.

$$x=\sqrt{13}+9 x=9-\sqrt{13}$$

Steps for Completing the Square

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$3x^{2}-54x=-204$$

Divide both sides by 3.

$$\frac{3x^{2}-54x}{3}=-\frac{204}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$x^{2}+\left(-\frac{54}{3}\right)x=-\frac{204}{3}$$

Divide -54 by 3.

$$x^{2}-18x=-\frac{204}{3}$$

Divide -204 by 3.

$$x^{2}-18x=-68$$

Divide -18, the coefficient of the x term, by 2 to get -9. Then add the square of -9 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-18x+\left(-9\right)^{2}=-68+\left(-9\right)^{2}$$

Square -9.

$$x^{2}-18x+81=-68+81$$

Add -68 to 81.

$$x^{2}-18x+81=13$$

Factor x^{2}-18x+81. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-9\right)^{2}=13$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-9\right)^{2}}=\sqrt{13}$$

Simplify.

$$x-9=\sqrt{13} x-9=-\sqrt{13}$$

Add 9 to both sides of the equation.

$$x=\sqrt{13}+9 x=9-\sqrt{13}$$