16x^2+16y^2-16x+24y-3=0
How do I solve this?
"Solving" means finding ordered pairs that satisfy the equation.
Your equation is a circle, and there are an infinite number of ordered pairs that will work.
Did you mean to put it in the standard form of a circle equation?
yes
16x^2+16y^2-16x+24y-3=0
16(x^2 - x + .....) + 16(y^2 + (3/2)y + ... = 3
16(x^2 - x + 1/4) + 16(y^2 + (3/2)y + 9/16) = 3 + 4 + 9
16(x-1/2)^2 + 16(y+3/4) = 16
(x-1/2)^2 + (y+3/4)^2 = 1
all vital info about the circle is now obvious.
thank you very much
The coordinate of all relative maxima 2x^4+x^3-33x^2-16x+16
To solve this equation, which represents a quadratic equation in two variables, you can follow these steps:
Step 1: Group the terms with x and y together:
16x^2 - 16x + 16y^2 + 24y - 3 = 0
Step 2: Rearrange the equation so that the x-terms and y-terms are separated:
16x^2 - 16x + 16y^2 + 24y = 3
Step 3: Complete the square for the x-terms:
To complete the square for the quadratic term, divide the coefficient of x by 2, square the result, and add it to both sides of the equation. In this case, the coefficient of x is -16, so the calculation would be:
(-16/2)^2 = 64
So, add 64 to both sides:
16x^2 - 16x + 64 + 16y^2 + 24y = 3 + 64
Step 4: Complete the square for the y-terms:
Perform the same process as in Step 3, but for the y-terms. The coefficient of y is 24, so the calculation would be:
(24/2)^2 = 144
So, add 144 to both sides:
16x^2 - 16x + 64 + 16y^2 + 24y + 144 = 3 + 64 + 144
Step 5: Simplify the equation:
Now simplify the equation by combining like terms:
16x^2 - 16x + 16y^2 + 24y + 208 = 211
Step 6: Factor the x and y terms:
Factor out the common factors from the x-terms and y-terms:
16(x^2 - x) + 16(y^2 + 3y) + 208 = 211
Step 7: Complete the square for the x-terms:
To complete the square for the quadratic term in the parentheses, divide the coefficient of x by 2, square the result, and add it inside the parentheses. In this case, the coefficient of x is -1, so the calculation would be:
(-1/2)^2 = 1/4
So, add 1/4 inside the parentheses:
16(x^2 - x + 1/4) + 16(y^2 + 3y) + 208 = 211
Step 8: Complete the square for the y-terms:
Perform the same process as in Step 7, but for the y-terms. The coefficient of y is 3, so the calculation would be:
(3/2)^2 = 9/4
So, add 9/4 inside the parentheses:
16(x^2 - x + 1/4) + 16(y^2 + 3y + 9/4) + 208 = 211
Step 9: Factor the completed squares:
Now, factor the completed squares:
16(x - 1/2)^2 + 16(y + 3/2)^2 + 208 = 211
Step 10: Simplify the equation:
Simplify the equation by subtracting 208 from both sides:
16(x - 1/2)^2 + 16(y + 3/2)^2 = 3
Step 11: Divide both sides by 16 to isolate the squared terms:
Divide both sides of the equation by 16:
(x - 1/2)^2 + (y + 3/2)^2/16 = 3/16
Step 12: The equation is now in standard form, which corresponds to the equation of a circle. By comparing the equation to the standard form of a circle equation:
(x - h)^2 + (y - k)^2 = r^2
we can identify the center point (h, k) and the radius r of the circle.
In this case:
Center point: (1/2, -3/2)
Radius: sqrt(3/16)
So, the solution to the equation is a circle with a center at (1/2, -3/2) and a radius of sqrt(3/16).