Solve 8-x^2+10-2left(x+1right)^2=0

$$8- { x }^{ 2 } +10-2 { \left(x+1 \right) }^{ 2 } =0$$

Solve for x

$$x = \frac{2 \sqrt{13} - 2}{3} \approx 1.737034184$$

$$x=\frac{-2\sqrt{13}-2}{3}\approx -3.070367517$$

Steps Using the Quadratic Formula

Add 8 and 10 to get 18.

$$18-x^{2}-2\left(x+1\right)^{2}=0$$

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

$$18-x^{2}-2\left(x^{2}+2x+1\right)=0$$

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

$$18-x^{2}-2x^{2}-4x-2=0$$

Combine -x^{2} and -2x^{2} to get -3x^{2}.

$$18-3x^{2}-4x-2=0$$

Subtract 2 from 18 to get 16.

$$16-3x^{2}-4x=0$$

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.

$$-3x^{2}-4x+16=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, -4 for b, and 16 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-3\right)\times 16}}{2\left(-3\right)}$$

Square -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\left(-3\right)\times 16}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-\left(-4\right)±\sqrt{16+12\times 16}}{2\left(-3\right)}$$

Multiply 12 times 16.

$$x=\frac{-\left(-4\right)±\sqrt{16+192}}{2\left(-3\right)}$$

Add 16 to 192.

$$x=\frac{-\left(-4\right)±\sqrt{208}}{2\left(-3\right)}$$

Take the square root of 208.

$$x=\frac{-\left(-4\right)±4\sqrt{13}}{2\left(-3\right)}$$

The opposite of -4 is 4.

$$x=\frac{4±4\sqrt{13}}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{4±4\sqrt{13}}{-6}$$

Now solve the equation x=\frac{4±4\sqrt{13}}{-6} when ± is plus. Add 4 to 4\sqrt{13}.

$$x=\frac{4\sqrt{13}+4}{-6}$$

Divide 4+4\sqrt{13} by -6.

$$x=\frac{-2\sqrt{13}-2}{3}$$

Now solve the equation x=\frac{4±4\sqrt{13}}{-6} when ± is minus. Subtract 4\sqrt{13} from 4.

$$x=\frac{4-4\sqrt{13}}{-6}$$

Divide 4-4\sqrt{13} by -6.

$$x=\frac{2\sqrt{13}-2}{3}$$

The equation is now solved.

$$x=\frac{-2\sqrt{13}-2}{3} x=\frac{2\sqrt{13}-2}{3}$$

Steps for Completing the Square

Add 8 and 10 to get 18.

$$18-x^{2}-2\left(x+1\right)^{2}=0$$

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

$$18-x^{2}-2\left(x^{2}+2x+1\right)=0$$

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

$$18-x^{2}-2x^{2}-4x-2=0$$

Combine -x^{2} and -2x^{2} to get -3x^{2}.

$$18-3x^{2}-4x-2=0$$

Subtract 2 from 18 to get 16.

$$16-3x^{2}-4x=0$$

Subtract 16 from both sides. Anything subtracted from zero gives its negation.

$$-3x^{2}-4x=-16$$

Divide both sides by -3.

$$\frac{-3x^{2}-4x}{-3}=-\frac{16}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$x^{2}+\left(-\frac{4}{-3}\right)x=-\frac{16}{-3}$$

Divide -4 by -3.

$$x^{2}+\frac{4}{3}x=-\frac{16}{-3}$$

Divide -16 by -3.

$$x^{2}+\frac{4}{3}x=\frac{16}{3}$$

Divide \frac{4}{3}, the coefficient of the x term, by 2 to get \frac{2}{3}. Then add the square of \frac{2}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{4}{3}x+\left(\frac{2}{3}\right)^{2}=\frac{16}{3}+\left(\frac{2}{3}\right)^{2}$$

Square \frac{2}{3} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{4}{3}x+\frac{4}{9}=\frac{16}{3}+\frac{4}{9}$$

Add \frac{16}{3} to \frac{4}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{4}{3}x+\frac{4}{9}=\frac{52}{9}$$

Factor x^{2}+\frac{4}{3}x+\frac{4}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{2}{3}\right)^{2}=\frac{52}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{2}{3}\right)^{2}}=\sqrt{\frac{52}{9}}$$

Simplify.

$$x+\frac{2}{3}=\frac{2\sqrt{13}}{3} x+\frac{2}{3}=-\frac{2\sqrt{13}}{3}$$

Subtract \frac{2}{3} from both sides of the equation.

$$x=\frac{2\sqrt{13}-2}{3} x=\frac{-2\sqrt{13}-2}{3}$$