Final answer:
To convert the given quadratic equation into standard form, complete the square for both x and y terms separately, then combine and rearrange the terms. The standard form of the given equation is (x + 1)^2 + 4(y + 3)^2 = 124.
Step-by-step explanation:
To convert the given equation x2 + 4y2 + 2x + 24y + 21 = 0 into standard form, we can complete the square for both x and y terms.
To complete the square for the x terms: x2 + 2x, add and subtract (2/2)2 = 1 inside the equationTo complete the square for the y terms: 4y2 + 24y, factor out the 4 from both terms and then add and subtract (
24/4/2)2 = 36 inside the equation.
The equation then becomes: x2 + 2x + 1 - 1 + 4(y2 + 6y + 36 - 36) + 21 = 0.
Rearrange and combine like terms to obtain the equation in standard form: (x + 1)2 + 4(y + 3)2 - 1 - 144 + 21 = 0, which simplifies to: (x + 1)2 + 4(y + 3)2 - 124 = 0.
Finally, add 124 to both sides to get the standard form: (x + 1)2 + 4(y + 3)2 = 124.