Solve (4-y)^2+y^2=100

$$( 4 - y ) ^ { 2 } + y ^ { 2 } = 100$$

Solve for y

$$y=\sqrt{46}+2\approx 8.782329983$$

$$y=2-\sqrt{46}\approx -4.782329983$$

Steps Using the Quadratic Formula

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

$$16-8y+y^{2}+y^{2}=100$$

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

$$16-8y+2y^{2}=100$$

Subtract 100 from both sides.

$$16-8y+2y^{2}-100=0$$

Subtract 100 from 16 to get -84.

$$-84-8y+2y^{2}=0$$

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.

$$2y^{2}-8y-84=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -8 for b, and -84 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 2\left(-84\right)}}{2\times 2}$$

Square -8.

$$y=\frac{-\left(-8\right)±\sqrt{64-4\times 2\left(-84\right)}}{2\times 2}$$

Multiply -4 times 2.

$$y=\frac{-\left(-8\right)±\sqrt{64-8\left(-84\right)}}{2\times 2}$$

Multiply -8 times -84.

$$y=\frac{-\left(-8\right)±\sqrt{64+672}}{2\times 2}$$

Add 64 to 672.

$$y=\frac{-\left(-8\right)±\sqrt{736}}{2\times 2}$$

Take the square root of 736.

$$y=\frac{-\left(-8\right)±4\sqrt{46}}{2\times 2}$$

The opposite of -8 is 8.

$$y=\frac{8±4\sqrt{46}}{2\times 2}$$

Multiply 2 times 2.

$$y=\frac{8±4\sqrt{46}}{4}$$

Now solve the equation y=\frac{8±4\sqrt{46}}{4} when ± is plus. Add 8 to 4\sqrt{46}.

$$y=\frac{4\sqrt{46}+8}{4}$$

Divide 8+4\sqrt{46} by 4.

$$y=\sqrt{46}+2$$

Now solve the equation y=\frac{8±4\sqrt{46}}{4} when ± is minus. Subtract 4\sqrt{46} from 8.

$$y=\frac{8-4\sqrt{46}}{4}$$

Divide 8-4\sqrt{46} by 4.

$$y=2-\sqrt{46}$$

The equation is now solved.

$$y=\sqrt{46}+2 y=2-\sqrt{46}$$

Steps for Completing the Square

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

$$16-8y+y^{2}+y^{2}=100$$

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

$$16-8y+2y^{2}=100$$

Subtract 16 from both sides.

$$-8y+2y^{2}=100-16$$

Subtract 16 from 100 to get 84.

$$-8y+2y^{2}=84$$

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.

$$2y^{2}-8y=84$$

Divide both sides by 2.

$$\frac{2y^{2}-8y}{2}=\frac{84}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$y^{2}+\left(-\frac{8}{2}\right)y=\frac{84}{2}$$

Divide -8 by 2.

$$y^{2}-4y=\frac{84}{2}$$

Divide 84 by 2.

$$y^{2}-4y=42$$

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.

$$y^{2}-4y+\left(-2\right)^{2}=42+\left(-2\right)^{2}$$

Square -2.

$$y^{2}-4y+4=42+4$$

Add 42 to 4.

$$y^{2}-4y+4=46$$

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

$$\left(y-2\right)^{2}=46$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-2\right)^{2}}=\sqrt{46}$$

Simplify.

$$y-2=\sqrt{46} y-2=-\sqrt{46}$$

Add 2 to both sides of the equation.

$$y=\sqrt{46}+2 y=2-\sqrt{46}$$