Solve x^2/6-1=(x+1)^2-1/5

$$\frac { x ^ { 2 } } { 6 } - 1 = ( x + 1 ) ^ { 2 } - \frac { 1 } { 5 }$$

Solve for x (complex solution)

$$x=\frac{-6+3\sqrt{2}i}{5}\approx -1.2+0.848528137i$$

$$x=\frac{-3\sqrt{2}i-6}{5}\approx -1.2-0.848528137i$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 30, the least common multiple of 6,5.

$$5x^{2}-30=30\left(x+1\right)^{2}-6$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.

$$5x^{2}-30=30\left(x^{2}+2x+1\right)-6$$

Use the distributive property to multiply 30 by x^{2}+2x+1.

$$5x^{2}-30=30x^{2}+60x+30-6$$

Subtract 6 from 30 to get 24.

$$5x^{2}-30=30x^{2}+60x+24$$

Subtract 30x^{2} from both sides.

$$5x^{2}-30-30x^{2}=60x+24$$

Combine 5x^{2} and -30x^{2} to get -25x^{2}.

$$-25x^{2}-30=60x+24$$

Subtract 60x from both sides.

$$-25x^{2}-30-60x=24$$

Subtract 24 from both sides.

$$-25x^{2}-30-60x-24=0$$

Subtract 24 from -30 to get -54.

$$-25x^{2}-54-60x=0$$

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.

$$-25x^{2}-60x-54=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -25 for a, -60 for b, and -54 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-60\right)±\sqrt{\left(-60\right)^{2}-4\left(-25\right)\left(-54\right)}}{2\left(-25\right)}$$

Square -60.

$$x=\frac{-\left(-60\right)±\sqrt{3600-4\left(-25\right)\left(-54\right)}}{2\left(-25\right)}$$

Multiply -4 times -25.

$$x=\frac{-\left(-60\right)±\sqrt{3600+100\left(-54\right)}}{2\left(-25\right)}$$

Multiply 100 times -54.

$$x=\frac{-\left(-60\right)±\sqrt{3600-5400}}{2\left(-25\right)}$$

Add 3600 to -5400.

$$x=\frac{-\left(-60\right)±\sqrt{-1800}}{2\left(-25\right)}$$

Take the square root of -1800.

$$x=\frac{-\left(-60\right)±30\sqrt{2}i}{2\left(-25\right)}$$

The opposite of -60 is 60.

$$x=\frac{60±30\sqrt{2}i}{2\left(-25\right)}$$

Multiply 2 times -25.

$$x=\frac{60±30\sqrt{2}i}{-50}$$

Now solve the equation x=\frac{60±30\sqrt{2}i}{-50} when ± is plus. Add 60 to 30i\sqrt{2}.

$$x=\frac{60+30\sqrt{2}i}{-50}$$

Divide 60+30i\sqrt{2} by -50.

$$x=\frac{-3\sqrt{2}i-6}{5}$$

Now solve the equation x=\frac{60±30\sqrt{2}i}{-50} when ± is minus. Subtract 30i\sqrt{2} from 60.

$$x=\frac{-30\sqrt{2}i+60}{-50}$$

Divide 60-30i\sqrt{2} by -50.

$$x=\frac{-6+3\sqrt{2}i}{5}$$

The equation is now solved.

$$x=\frac{-3\sqrt{2}i-6}{5} x=\frac{-6+3\sqrt{2}i}{5}$$

Steps for Completing the Square

Multiply both sides of the equation by 30, the least common multiple of 6,5.

$$5x^{2}-30=30\left(x+1\right)^{2}-6$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.

$$5x^{2}-30=30\left(x^{2}+2x+1\right)-6$$

Use the distributive property to multiply 30 by x^{2}+2x+1.

$$5x^{2}-30=30x^{2}+60x+30-6$$

Subtract 6 from 30 to get 24.

$$5x^{2}-30=30x^{2}+60x+24$$

Subtract 30x^{2} from both sides.

$$5x^{2}-30-30x^{2}=60x+24$$

Combine 5x^{2} and -30x^{2} to get -25x^{2}.

$$-25x^{2}-30=60x+24$$

Subtract 60x from both sides.

$$-25x^{2}-30-60x=24$$

Add 30 to both sides.

$$-25x^{2}-60x=24+30$$

Add 24 and 30 to get 54.

$$-25x^{2}-60x=54$$

Divide both sides by -25.

$$\frac{-25x^{2}-60x}{-25}=\frac{54}{-25}$$

Dividing by -25 undoes the multiplication by -25.

$$x^{2}+\left(-\frac{60}{-25}\right)x=\frac{54}{-25}$$

Reduce the fraction \frac{-60}{-25} to lowest terms by extracting and canceling out 5.

$$x^{2}+\frac{12}{5}x=\frac{54}{-25}$$

Divide 54 by -25.

$$x^{2}+\frac{12}{5}x=-\frac{54}{25}$$

Divide \frac{12}{5}, the coefficient of the x term, by 2 to get \frac{6}{5}. Then add the square of \frac{6}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{12}{5}x+\left(\frac{6}{5}\right)^{2}=-\frac{54}{25}+\left(\frac{6}{5}\right)^{2}$$

Square \frac{6}{5} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{12}{5}x+\frac{36}{25}=\frac{-54+36}{25}$$

Add -\frac{54}{25} to \frac{36}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{12}{5}x+\frac{36}{25}=-\frac{18}{25}$$

Factor x^{2}+\frac{12}{5}x+\frac{36}{25}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{6}{5}\right)^{2}=-\frac{18}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{6}{5}\right)^{2}}=\sqrt{-\frac{18}{25}}$$

Simplify.

$$x+\frac{6}{5}=\frac{3\sqrt{2}i}{5} x+\frac{6}{5}=-\frac{3\sqrt{2}i}{5}$$

Subtract \frac{6}{5} from both sides of the equation.

$$x=\frac{-6+3\sqrt{2}i}{5} x=\frac{-3\sqrt{2}i-6}{5}$$