Solve x^2+6/2-8-x/10=1

$$\frac { x ^ { 2 } + 6 } { 2 } - \frac { 8 - x } { 10 } = 1$$

Solve for x (complex solution)

$$x=\frac{-1+\sqrt{239}i}{10}\approx -0.1+1.545962483i$$

$$x=\frac{-\sqrt{239}i-1}{10}\approx -0.1-1.545962483i$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 10, the least common multiple of 2,10.

$$5\left(x^{2}+6\right)-\left(8-x\right)=10$$

Use the distributive property to multiply 5 by x^{2}+6.

$$5x^{2}+30-\left(8-x\right)=10$$

To find the opposite of 8-x, find the opposite of each term.

$$5x^{2}+30-8+x=10$$

Subtract 8 from 30 to get 22.

$$5x^{2}+22+x=10$$

Subtract 10 from both sides.

$$5x^{2}+22+x-10=0$$

Subtract 10 from 22 to get 12.

$$5x^{2}+12+x=0$$

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.

$$5x^{2}+x+12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, 1 for b, and 12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1±\sqrt{1^{2}-4\times 5\times 12}}{2\times 5}$$

Square 1.

$$x=\frac{-1±\sqrt{1-4\times 5\times 12}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-1±\sqrt{1-20\times 12}}{2\times 5}$$

Multiply -20 times 12.

$$x=\frac{-1±\sqrt{1-240}}{2\times 5}$$

Add 1 to -240.

$$x=\frac{-1±\sqrt{-239}}{2\times 5}$$

Take the square root of -239.

$$x=\frac{-1±\sqrt{239}i}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{-1±\sqrt{239}i}{10}$$

Now solve the equation x=\frac{-1±\sqrt{239}i}{10} when ± is plus. Add -1 to i\sqrt{239}.

$$x=\frac{-1+\sqrt{239}i}{10}$$

Now solve the equation x=\frac{-1±\sqrt{239}i}{10} when ± is minus. Subtract i\sqrt{239} from -1.

$$x=\frac{-\sqrt{239}i-1}{10}$$

The equation is now solved.

$$x=\frac{-1+\sqrt{239}i}{10} x=\frac{-\sqrt{239}i-1}{10}$$

Steps for Completing the Square

Multiply both sides of the equation by 10, the least common multiple of 2,10.

$$5\left(x^{2}+6\right)-\left(8-x\right)=10$$

Use the distributive property to multiply 5 by x^{2}+6.

$$5x^{2}+30-\left(8-x\right)=10$$

To find the opposite of 8-x, find the opposite of each term.

$$5x^{2}+30-8+x=10$$

Subtract 8 from 30 to get 22.

$$5x^{2}+22+x=10$$

Subtract 22 from both sides.

$$5x^{2}+x=10-22$$

Subtract 22 from 10 to get -12.

$$5x^{2}+x=-12$$

Divide both sides by 5.

$$\frac{5x^{2}+x}{5}=-\frac{12}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$x^{2}+\frac{1}{5}x=-\frac{12}{5}$$

Divide \frac{1}{5}, the coefficient of the x term, by 2 to get \frac{1}{10}. Then add the square of \frac{1}{10} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{1}{5}x+\left(\frac{1}{10}\right)^{2}=-\frac{12}{5}+\left(\frac{1}{10}\right)^{2}$$

Square \frac{1}{10} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{1}{5}x+\frac{1}{100}=-\frac{12}{5}+\frac{1}{100}$$

Add -\frac{12}{5} to \frac{1}{100} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{1}{5}x+\frac{1}{100}=-\frac{239}{100}$$

Factor x^{2}+\frac{1}{5}x+\frac{1}{100}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{10}\right)^{2}=-\frac{239}{100}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{10}\right)^{2}}=\sqrt{-\frac{239}{100}}$$

Simplify.

$$x+\frac{1}{10}=\frac{\sqrt{239}i}{10} x+\frac{1}{10}=-\frac{\sqrt{239}i}{10}$$

Subtract \frac{1}{10} from both sides of the equation.

$$x=\frac{-1+\sqrt{239}i}{10} x=\frac{-\sqrt{239}i-1}{10}$$