Solve x^2+left(4+xright)^2=36

$${ x }^{ 2 } + { \left(4+x \right) }^{ 2 } =36$$

Solve for x (complex solution)

$$x=\sqrt{14}-2\approx 1.741657387$$

$$x=-\left(\sqrt{14}+2\right)\approx -5.741657387$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(4+x\right)^{2}.

$$x^{2}+16+8x+x^{2}=36$$

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

$$2x^{2}+16+8x=36$$

Subtract 36 from both sides.

$$2x^{2}+16+8x-36=0$$

Subtract 36 from 16 to get -20.

$$2x^{2}-20+8x=0$$

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.

$$2x^{2}+8x-20=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 8 for b, and -20 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-8±\sqrt{8^{2}-4\times 2\left(-20\right)}}{2\times 2}$$

Square 8.

$$x=\frac{-8±\sqrt{64-4\times 2\left(-20\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-8±\sqrt{64-8\left(-20\right)}}{2\times 2}$$

Multiply -8 times -20.

$$x=\frac{-8±\sqrt{64+160}}{2\times 2}$$

Add 64 to 160.

$$x=\frac{-8±\sqrt{224}}{2\times 2}$$

Take the square root of 224.

$$x=\frac{-8±4\sqrt{14}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-8±4\sqrt{14}}{4}$$

Now solve the equation x=\frac{-8±4\sqrt{14}}{4} when ± is plus. Add -8 to 4\sqrt{14}.

$$x=\frac{4\sqrt{14}-8}{4}$$

Divide -8+4\sqrt{14} by 4.

$$x=\sqrt{14}-2$$

Now solve the equation x=\frac{-8±4\sqrt{14}}{4} when ± is minus. Subtract 4\sqrt{14} from -8.

$$x=\frac{-4\sqrt{14}-8}{4}$$

Divide -8-4\sqrt{14} by 4.

$$x=-\sqrt{14}-2$$

The equation is now solved.

$$x=\sqrt{14}-2 x=-\sqrt{14}-2$$

Steps for Completing the Square

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(4+x\right)^{2}.

$$x^{2}+16+8x+x^{2}=36$$

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

$$2x^{2}+16+8x=36$$

Subtract 16 from both sides.

$$2x^{2}+8x=36-16$$

Subtract 16 from 36 to get 20.

$$2x^{2}+8x=20$$

Divide both sides by 2.

$$\frac{2x^{2}+8x}{2}=\frac{20}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{8}{2}x=\frac{20}{2}$$

Divide 8 by 2.

$$x^{2}+4x=\frac{20}{2}$$

Divide 20 by 2.

$$x^{2}+4x=10$$

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.

$$x^{2}+4x+2^{2}=10+2^{2}$$

Square 2.

$$x^{2}+4x+4=10+4$$

Add 10 to 4.

$$x^{2}+4x+4=14$$

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}.

$$\left(x+2\right)^{2}=14$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+2\right)^{2}}=\sqrt{14}$$

Simplify.

$$x+2=\sqrt{14} x+2=-\sqrt{14}$$

Subtract 2 from both sides of the equation.

$$x=\sqrt{14}-2 x=-\sqrt{14}-2$$

Solve for x

$$x=\sqrt{14}-2\approx 1.741657387$$

$$x=-\sqrt{14}-2\approx -5.741657387$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(4+x\right)^{2}.

$$x^{2}+16+8x+x^{2}=36$$

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

$$2x^{2}+16+8x=36$$

Subtract 36 from both sides.

$$2x^{2}+16+8x-36=0$$

Subtract 36 from 16 to get -20.

$$2x^{2}-20+8x=0$$

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.

$$2x^{2}+8x-20=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 8 for b, and -20 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-8±\sqrt{8^{2}-4\times 2\left(-20\right)}}{2\times 2}$$

Square 8.

$$x=\frac{-8±\sqrt{64-4\times 2\left(-20\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-8±\sqrt{64-8\left(-20\right)}}{2\times 2}$$

Multiply -8 times -20.

$$x=\frac{-8±\sqrt{64+160}}{2\times 2}$$

Add 64 to 160.

$$x=\frac{-8±\sqrt{224}}{2\times 2}$$

Take the square root of 224.

$$x=\frac{-8±4\sqrt{14}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-8±4\sqrt{14}}{4}$$

Now solve the equation x=\frac{-8±4\sqrt{14}}{4} when ± is plus. Add -8 to 4\sqrt{14}.

$$x=\frac{4\sqrt{14}-8}{4}$$

Divide -8+4\sqrt{14} by 4.

$$x=\sqrt{14}-2$$

Now solve the equation x=\frac{-8±4\sqrt{14}}{4} when ± is minus. Subtract 4\sqrt{14} from -8.

$$x=\frac{-4\sqrt{14}-8}{4}$$

Divide -8-4\sqrt{14} by 4.

$$x=-\sqrt{14}-2$$

The equation is now solved.

$$x=\sqrt{14}-2 x=-\sqrt{14}-2$$

Steps for Completing the Square

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(4+x\right)^{2}.

$$x^{2}+16+8x+x^{2}=36$$

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

$$2x^{2}+16+8x=36$$

Subtract 16 from both sides.

$$2x^{2}+8x=36-16$$

Subtract 16 from 36 to get 20.

$$2x^{2}+8x=20$$

Divide both sides by 2.

$$\frac{2x^{2}+8x}{2}=\frac{20}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{8}{2}x=\frac{20}{2}$$

Divide 8 by 2.

$$x^{2}+4x=\frac{20}{2}$$

Divide 20 by 2.

$$x^{2}+4x=10$$

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.

$$x^{2}+4x+2^{2}=10+2^{2}$$

Square 2.

$$x^{2}+4x+4=10+4$$

Add 10 to 4.

$$x^{2}+4x+4=14$$

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}.

$$\left(x+2\right)^{2}=14$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+2\right)^{2}}=\sqrt{14}$$

Simplify.

$$x+2=\sqrt{14} x+2=-\sqrt{14}$$

Subtract 2 from both sides of the equation.

$$x=\sqrt{14}-2 x=-\sqrt{14}-2$$