Solve 1/2x^2-pi/2x+57/8-1=0

$$\frac { 1 } { 2 } x ^ { 2 } - \frac { \pi } { 2 } x + \frac { 57 } { 8 } - 1 = 0$$

Solve for x (complex solution)

$$x=\frac{\pi +i\sqrt{-\pi ^{2}+49}}{2}\approx 1.570796327+3.127714645i$$

$$x=\frac{-i\sqrt{-\pi ^{2}+49}+\pi }{2}\approx 1.570796327-3.127714645i$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 8, the least common multiple of 2,8.

$$8\left(\frac{1}{2}x^{2}-\frac{\pi }{2}x\right)+57-8=0$$

Express \frac{\pi }{2}x as a single fraction.

$$8\left(\frac{1}{2}x^{2}-\frac{\pi x}{2}\right)+57-8=0$$

Use the distributive property to multiply 8 by \frac{1}{2}x^{2}-\frac{\pi x}{2}.

$$4x^{2}+8\left(-\frac{\pi x}{2}\right)+57-8=0$$

Cancel out 2, the greatest common factor in 8 and 2.

$$4x^{2}-4\pi x+57-8=0$$

Subtract 8 from 57 to get 49.

$$4x^{2}-4\pi x+49=0$$

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.

$$4x^{2}+\left(-4\pi \right)x+49=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -4\pi for b, and 49 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4\pi \right)±\sqrt{\left(-4\pi \right)^{2}-4\times 4\times 49}}{2\times 4}$$

Square -4\pi .

$$x=\frac{-\left(-4\pi \right)±\sqrt{16\pi ^{2}-4\times 4\times 49}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-4\pi \right)±\sqrt{16\pi ^{2}-16\times 49}}{2\times 4}$$

Multiply -16 times 49.

$$x=\frac{-\left(-4\pi \right)±\sqrt{16\pi ^{2}-784}}{2\times 4}$$

Take the square root of 16\pi ^{2}-784.

$$x=\frac{-\left(-4\pi \right)±4i\sqrt{-\pi ^{2}+49}}{2\times 4}$$

The opposite of -4\pi is 4\pi .

$$x=\frac{4\pi ±4i\sqrt{-\pi ^{2}+49}}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{4\pi ±4i\sqrt{-\pi ^{2}+49}}{8}$$

Now solve the equation x=\frac{4\pi ±4i\sqrt{-\pi ^{2}+49}}{8} when ± is plus. Add 4\pi to 4i\sqrt{-\pi ^{2}+49}.

$$x=\frac{4\pi +4i\sqrt{-\pi ^{2}+49}}{8}$$

Divide 4\pi +4i\sqrt{49-\pi ^{2}} by 8.

$$x=\frac{\pi +i\sqrt{-\pi ^{2}+49}}{2}$$

Now solve the equation x=\frac{4\pi ±4i\sqrt{-\pi ^{2}+49}}{8} when ± is minus. Subtract 4i\sqrt{-\pi ^{2}+49} from 4\pi .

$$x=\frac{-4i\sqrt{-\pi ^{2}+49}+4\pi }{8}$$

Divide 4\pi -4i\sqrt{49-\pi ^{2}} by 8.

$$x=\frac{-i\sqrt{-\pi ^{2}+49}+\pi }{2}$$

The equation is now solved.

$$x=\frac{\pi +i\sqrt{-\pi ^{2}+49}}{2} x=\frac{-i\sqrt{-\pi ^{2}+49}+\pi }{2}$$

Steps for Completing the Square

Multiply both sides of the equation by 8, the least common multiple of 2,8.

$$8\left(\frac{1}{2}x^{2}-\frac{\pi }{2}x\right)+57-8=0$$

Express \frac{\pi }{2}x as a single fraction.

$$8\left(\frac{1}{2}x^{2}-\frac{\pi x}{2}\right)+57-8=0$$

Use the distributive property to multiply 8 by \frac{1}{2}x^{2}-\frac{\pi x}{2}.

$$4x^{2}+8\left(-\frac{\pi x}{2}\right)+57-8=0$$

Cancel out 2, the greatest common factor in 8 and 2.

$$4x^{2}-4\pi x+57-8=0$$

Subtract 8 from 57 to get 49.

$$4x^{2}-4\pi x+49=0$$

Subtract 49 from both sides. Anything subtracted from zero gives its negation.

$$4x^{2}-4\pi x=-49$$

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.

$$4x^{2}+\left(-4\pi \right)x=-49$$

Divide both sides by 4.

$$\frac{4x^{2}+\left(-4\pi \right)x}{4}=-\frac{49}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\left(-\frac{4\pi }{4}\right)x=-\frac{49}{4}$$

Divide -4\pi by 4.

$$x^{2}+\left(-\pi \right)x=-\frac{49}{4}$$

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

$$x^{2}+\left(-\pi \right)x+\left(-\frac{\pi }{2}\right)^{2}=-\frac{49}{4}+\left(-\frac{\pi }{2}\right)^{2}$$

Square -\frac{\pi }{2}.

$$x^{2}+\left(-\pi \right)x+\frac{\pi ^{2}}{4}=\frac{-49+\pi ^{2}}{4}$$

Add -\frac{49}{4} to \frac{\pi ^{2}}{4}.

$$x^{2}+\left(-\pi \right)x+\frac{\pi ^{2}}{4}=\frac{\pi ^{2}-49}{4}$$

Factor x^{2}+\left(-\pi \right)x+\frac{\pi ^{2}}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{\pi }{2}\right)^{2}=\frac{\pi ^{2}-49}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{\pi }{2}\right)^{2}}=\sqrt{\frac{\pi ^{2}-49}{4}}$$

Simplify.

$$x-\frac{\pi }{2}=\frac{i\sqrt{-\pi ^{2}+49}}{2} x-\frac{\pi }{2}=-\frac{i\sqrt{-\pi ^{2}+49}}{2}$$

Add \frac{\pi }{2} to both sides of the equation.

$$x=\frac{\pi +i\sqrt{-\pi ^{2}+49}}{2} x=\frac{-i\sqrt{-\pi ^{2}+49}+\pi }{2}$$