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PLEASE HELP ASAP!! Complete the square to rewrite the following equation. Identify the center and radius of the circle. You must show all work and calculations to receive credit.

x2 + 2x + y2 + 4y = 20


Sagot :

Answer:

(i) Center = (-1, -2)

(ii) Radius = 5 units

General Circle Equation:

  • (x - h)² + (y - k)² = r²

Where:

  • (h, k) is the center points
  • r denotes the radius

Rewriting the equation:

[tex]\sf x^2 + 2x + y^2+ 4y = 20[/tex]

[tex]\sf x^2 + 2x + 1^2 - 1^2 + y^2 + 4y + 2^2 - 2^2 = 20[/tex]

[tex]\sf (x + 1)^2 - 1 + (y + 2)^2 -4 = 20[/tex]

[tex]\sf (x + 1)^2 + (y + 2)^2 = 20 + 5[/tex]

[tex]\sf (x + 1)^2 + (y + 2)^2 = 25[/tex]

[tex]\sf (x -(- 1))^2 + (y -(- 2))^2 = 5^2 \quad \leftarrow \ \bf General \ Circle \ Equation[/tex]

Identify the following:

  • (h, k) = (-1, -2), radius = 5 units

Answer:

Completing the square:  Circles

Add the square of half the coefficients of both first degree terms (x and y) to both sides:

[tex]\begin{aligned}\implies x^2+2x+\left(\dfrac{2}{2}\right)^2+y^2+4y+\left(\dfrac{4}{2}\right)^2 & =20+\left(\dfrac{2}{2}\right)^2+\left(\dfrac{4}{2}\right)^2\\\\x^2+2x+1+y^2+4y+4 & = 20+1+4\\\\x^2+2x+1+y^2+4y+4 & = 25\end{aligned}[/tex]

Factor the two trinomials on the left side of the equation:

[tex]\begin{aligned} \implies x^2+2x+1+y^2+4y+4 & = 25\\\\ \implies (x+1)^2+(y+2)^2 & = 25 \end{aligned}[/tex]

Equation of a circle:  [tex](x-a)^2+(y-b)^2=r^2[/tex]

(where (a, b) is the center and r is the radius)

Comparing constants:

[tex]\displaystyle (x-a)^2+(y-b)^2=r^2\\\\\phantom{(((((}\downarrow \phantom{(((((((((} \downarrow \phantom{(((((} \downarrow \\\\(x+1)^2+(y+2)^2=25[/tex]

Therefore:

[tex]-a=1 \implies a=-1[/tex]

[tex]-b=2 \implies b=-2[/tex]

[tex]r^2=25 \implies r=\sqrt{25}=5[/tex]

Conclusion:

  • center = (-1, -2)
  • radius = 5 units
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