Solve 8x+2-3x^2+1=35-x^2

$$8 x + 2 - 3 x ^ { 2 } + 1 = 35 - x ^ { 2 }$$

Solve for x (complex solution)

$$x=2+2\sqrt{3}i\approx 2+3.464101615i$$

$$x=-2\sqrt{3}i+2\approx 2-3.464101615i$$

Steps Using the Quadratic Formula

Add 2 and 1 to get 3.

$$8x+3-3x^{2}=35-x^{2}$$

Subtract 35 from both sides.

$$8x+3-3x^{2}-35=-x^{2}$$

Subtract 35 from 3 to get -32.

$$8x-32-3x^{2}=-x^{2}$$

Add x^{2} to both sides.

$$8x-32-3x^{2}+x^{2}=0$$

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

$$8x-32-2x^{2}=0$$

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.

$$-2x^{2}+8x-32=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 8 for b, and -32 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-8±\sqrt{8^{2}-4\left(-2\right)\left(-32\right)}}{2\left(-2\right)}$$

Square 8.

$$x=\frac{-8±\sqrt{64-4\left(-2\right)\left(-32\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-8±\sqrt{64+8\left(-32\right)}}{2\left(-2\right)}$$

Multiply 8 times -32.

$$x=\frac{-8±\sqrt{64-256}}{2\left(-2\right)}$$

Add 64 to -256.

$$x=\frac{-8±\sqrt{-192}}{2\left(-2\right)}$$

Take the square root of -192.

$$x=\frac{-8±8\sqrt{3}i}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-8±8\sqrt{3}i}{-4}$$

Now solve the equation x=\frac{-8±8\sqrt{3}i}{-4} when ± is plus. Add -8 to 8i\sqrt{3}.

$$x=\frac{-8+8\sqrt{3}i}{-4}$$

Divide -8+8i\sqrt{3} by -4.

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

Now solve the equation x=\frac{-8±8\sqrt{3}i}{-4} when ± is minus. Subtract 8i\sqrt{3} from -8.

$$x=\frac{-8\sqrt{3}i-8}{-4}$$

Divide -8-8i\sqrt{3} by -4.

$$x=2+2\sqrt{3}i$$

The equation is now solved.

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

Steps for Completing the Square

Add 2 and 1 to get 3.

$$8x+3-3x^{2}=35-x^{2}$$

Add x^{2} to both sides.

$$8x+3-3x^{2}+x^{2}=35$$

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

$$8x+3-2x^{2}=35$$

Subtract 3 from both sides.

$$8x-2x^{2}=35-3$$

Subtract 3 from 35 to get 32.

$$8x-2x^{2}=32$$

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.

$$-2x^{2}+8x=32$$

Divide both sides by -2.

$$\frac{-2x^{2}+8x}{-2}=\frac{32}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{8}{-2}x=\frac{32}{-2}$$

Divide 8 by -2.

$$x^{2}-4x=\frac{32}{-2}$$

Divide 32 by -2.

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

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

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

Square -2.

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

Add -16 to 4.

$$x^{2}-4x+4=-12$$

Factor x^{2}-4x+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}.

$$\left(x-2\right)^{2}=-12$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{-12}$$

Simplify.

$$x-2=2\sqrt{3}i x-2=-2\sqrt{3}i$$

Add 2 to both sides of the equation.

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