Solve 2(x^2-9)/5-x+1/2=x-41/4

$$\frac { 2 ( x ^ { 2 } - 9 ) } { 5 } - \frac { x + 1 } { 2 } = \frac { x - 41 } { 4 }$$

Solve for x (complex solution)

$$x=\frac{15+\sqrt{3711}i}{16}\approx 0.9375+3.807373603i$$

$$x=\frac{-\sqrt{3711}i+15}{16}\approx 0.9375-3.807373603i$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 20, the least common multiple of 5,2,4.

$$4\times 2\left(x^{2}-9\right)-10\left(x+1\right)=5\left(x-41\right)$$

Multiply 4 and 2 to get 8.

$$8\left(x^{2}-9\right)-10\left(x+1\right)=5\left(x-41\right)$$

Use the distributive property to multiply 8 by x^{2}-9.

$$8x^{2}-72-10\left(x+1\right)=5\left(x-41\right)$$

Use the distributive property to multiply -10 by x+1.

$$8x^{2}-72-10x-10=5\left(x-41\right)$$

Subtract 10 from -72 to get -82.

$$8x^{2}-82-10x=5\left(x-41\right)$$

Use the distributive property to multiply 5 by x-41.

$$8x^{2}-82-10x=5x-205$$

Subtract 5x from both sides.

$$8x^{2}-82-10x-5x=-205$$

Combine -10x and -5x to get -15x.

$$8x^{2}-82-15x=-205$$

Add 205 to both sides.

$$8x^{2}-82-15x+205=0$$

Add -82 and 205 to get 123.

$$8x^{2}+123-15x=0$$

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.

$$8x^{2}-15x+123=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 8 for a, -15 for b, and 123 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-15\right)±\sqrt{\left(-15\right)^{2}-4\times 8\times 123}}{2\times 8}$$

Square -15.

$$x=\frac{-\left(-15\right)±\sqrt{225-4\times 8\times 123}}{2\times 8}$$

Multiply -4 times 8.

$$x=\frac{-\left(-15\right)±\sqrt{225-32\times 123}}{2\times 8}$$

Multiply -32 times 123.

$$x=\frac{-\left(-15\right)±\sqrt{225-3936}}{2\times 8}$$

Add 225 to -3936.

$$x=\frac{-\left(-15\right)±\sqrt{-3711}}{2\times 8}$$

Take the square root of -3711.

$$x=\frac{-\left(-15\right)±\sqrt{3711}i}{2\times 8}$$

The opposite of -15 is 15.

$$x=\frac{15±\sqrt{3711}i}{2\times 8}$$

Multiply 2 times 8.

$$x=\frac{15±\sqrt{3711}i}{16}$$

Now solve the equation x=\frac{15±\sqrt{3711}i}{16} when ± is plus. Add 15 to i\sqrt{3711}.

$$x=\frac{15+\sqrt{3711}i}{16}$$

Now solve the equation x=\frac{15±\sqrt{3711}i}{16} when ± is minus. Subtract i\sqrt{3711} from 15.

$$x=\frac{-\sqrt{3711}i+15}{16}$$

The equation is now solved.

$$x=\frac{15+\sqrt{3711}i}{16} x=\frac{-\sqrt{3711}i+15}{16}$$

Steps for Completing the Square

Multiply both sides of the equation by 20, the least common multiple of 5,2,4.

$$4\times 2\left(x^{2}-9\right)-10\left(x+1\right)=5\left(x-41\right)$$

Multiply 4 and 2 to get 8.

$$8\left(x^{2}-9\right)-10\left(x+1\right)=5\left(x-41\right)$$

Use the distributive property to multiply 8 by x^{2}-9.

$$8x^{2}-72-10\left(x+1\right)=5\left(x-41\right)$$

Use the distributive property to multiply -10 by x+1.

$$8x^{2}-72-10x-10=5\left(x-41\right)$$

Subtract 10 from -72 to get -82.

$$8x^{2}-82-10x=5\left(x-41\right)$$

Use the distributive property to multiply 5 by x-41.

$$8x^{2}-82-10x=5x-205$$

Subtract 5x from both sides.

$$8x^{2}-82-10x-5x=-205$$

Combine -10x and -5x to get -15x.

$$8x^{2}-82-15x=-205$$

Add 82 to both sides.

$$8x^{2}-15x=-205+82$$

Add -205 and 82 to get -123.

$$8x^{2}-15x=-123$$

Divide both sides by 8.

$$\frac{8x^{2}-15x}{8}=-\frac{123}{8}$$

Dividing by 8 undoes the multiplication by 8.

$$x^{2}-\frac{15}{8}x=-\frac{123}{8}$$

Divide -\frac{15}{8}, the coefficient of the x term, by 2 to get -\frac{15}{16}. Then add the square of -\frac{15}{16} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{15}{8}x+\left(-\frac{15}{16}\right)^{2}=-\frac{123}{8}+\left(-\frac{15}{16}\right)^{2}$$

Square -\frac{15}{16} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{15}{8}x+\frac{225}{256}=-\frac{123}{8}+\frac{225}{256}$$

Add -\frac{123}{8} to \frac{225}{256} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{15}{8}x+\frac{225}{256}=-\frac{3711}{256}$$

Factor x^{2}-\frac{15}{8}x+\frac{225}{256}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{15}{16}\right)^{2}=-\frac{3711}{256}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{15}{16}\right)^{2}}=\sqrt{-\frac{3711}{256}}$$

Simplify.

$$x-\frac{15}{16}=\frac{\sqrt{3711}i}{16} x-\frac{15}{16}=-\frac{\sqrt{3711}i}{16}$$

Add \frac{15}{16} to both sides of the equation.

$$x=\frac{15+\sqrt{3711}i}{16} x=\frac{-\sqrt{3711}i+15}{16}$$