Solve 6x^2-pix+3pi=0

$$6 x ^ { 2 } - \pi x + 3 \pi = 0$$

Solve for x (complex solution)

$$x=\frac{\pi +i\sqrt{-\pi ^{2}+72\pi }}{12}\approx 0.261799388+1.225666108i$$

$$x=\frac{-i\sqrt{-\pi ^{2}+72\pi }+\pi }{12}\approx 0.261799388-1.225666108i$$

Steps Using the Quadratic Formula

Reorder the terms.

$$6x^{2}-\pi x+3\pi =0$$

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.

$$6x^{2}+\left(-\pi \right)x+3\pi =0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, -\pi for b, and 3\pi for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\pi \right)±\sqrt{\left(-\pi \right)^{2}-4\times 6\times 3\pi }}{2\times 6}$$

Square -\pi .

$$x=\frac{-\left(-\pi \right)±\sqrt{\pi ^{2}-4\times 6\times 3\pi }}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-\left(-\pi \right)±\sqrt{\pi ^{2}-24\times 3\pi }}{2\times 6}$$

Multiply -24 times 3\pi .

$$x=\frac{-\left(-\pi \right)±\sqrt{\pi ^{2}-72\pi }}{2\times 6}$$

Add \pi ^{2} to -72\pi .

$$x=\frac{-\left(-\pi \right)±\sqrt{\pi \left(\pi -72\right)}}{2\times 6}$$

Take the square root of \pi \left(\pi -72\right).

$$x=\frac{-\left(-\pi \right)±i\sqrt{\pi \left(72-\pi \right)}}{2\times 6}$$

The opposite of -\pi is \pi .

$$x=\frac{\pi ±i\sqrt{\pi \left(72-\pi \right)}}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{\pi ±i\sqrt{\pi \left(72-\pi \right)}}{12}$$

Now solve the equation x=\frac{\pi ±i\sqrt{\pi \left(72-\pi \right)}}{12} when ± is plus. Add \pi to i\sqrt{\pi \left(-\pi +72\right)}.

$$x=\frac{\pi +i\sqrt{\pi \left(72-\pi \right)}}{12}$$

Now solve the equation x=\frac{\pi ±i\sqrt{\pi \left(72-\pi \right)}}{12} when ± is minus. Subtract i\sqrt{\pi \left(-\pi +72\right)} from \pi .

$$x=\frac{-i\sqrt{\pi \left(72-\pi \right)}+\pi }{12}$$

The equation is now solved.

$$x=\frac{\pi +i\sqrt{\pi \left(72-\pi \right)}}{12} x=\frac{-i\sqrt{\pi \left(72-\pi \right)}+\pi }{12}$$

Steps for Completing the Square

Subtract 3\pi from both sides. Anything subtracted from zero gives its negation.

$$6x^{2}-\pi x=-3\pi$$

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.

$$6x^{2}+\left(-\pi \right)x=-3\pi$$

Divide both sides by 6.

$$\frac{6x^{2}+\left(-\pi \right)x}{6}=-\frac{3\pi }{6}$$

Dividing by 6 undoes the multiplication by 6.

$$x^{2}+\left(-\frac{\pi }{6}\right)x=-\frac{3\pi }{6}$$

Divide -3\pi by 6.

$$x^{2}+\left(-\frac{\pi }{6}\right)x=-\frac{\pi }{2}$$

Divide -\frac{\pi }{6}, the coefficient of the x term, by 2 to get -\frac{\pi }{12}. Then add the square of -\frac{\pi }{12} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\left(-\frac{\pi }{6}\right)x+\left(-\frac{\pi }{12}\right)^{2}=-\frac{\pi }{2}+\left(-\frac{\pi }{12}\right)^{2}$$

Square -\frac{\pi }{12}.

$$x^{2}+\left(-\frac{\pi }{6}\right)x+\frac{\pi ^{2}}{144}=-\frac{\pi }{2}+\frac{\pi ^{2}}{144}$$

Add -\frac{\pi }{2} to \frac{\pi ^{2}}{144}.

$$x^{2}+\left(-\frac{\pi }{6}\right)x+\frac{\pi ^{2}}{144}=\frac{\pi \left(\pi -72\right)}{144}$$

Factor x^{2}+\left(-\frac{\pi }{6}\right)x+\frac{\pi ^{2}}{144}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{\pi }{12}\right)^{2}=\frac{\pi \left(\pi -72\right)}{144}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{\pi }{12}\right)^{2}}=\sqrt{\frac{\pi \left(\pi -72\right)}{144}}$$

Simplify.

$$x-\frac{\pi }{12}=\frac{i\sqrt{\pi \left(72-\pi \right)}}{12} x-\frac{\pi }{12}=-\frac{i\sqrt{\pi \left(72-\pi \right)}}{12}$$

Add \frac{\pi }{12} to both sides of the equation.

$$x=\frac{\pi +i\sqrt{\pi \left(72-\pi \right)}}{12} x=\frac{-i\sqrt{\pi \left(72-\pi \right)}+\pi }{12}$$