What is the standard form of 4x²+4y²-20x+12y-2=0
4x²+4y²-20x+12y-2=0
4x²-20x+4y²+12y = 2
4(x²-5x) + 4(y²+3y) = 2
4(x²-5x+25/4) + 4(y²+3y+9/4) = 2 + 4*25/4 + 4*9/4
4(x-5/2)² + 4(y+3/2)² = 36
(x-5/2)² + (y+3/2)² = 9
why is it nine. It should be =6
Thanks for this solution it helps me lot heheehag
4x²+4y²+12y-7=0
4X²+4Y²+12Y-7=7
Well, let me crack a little joke for you while solving this equation!
Why was the math book sad? Because it had too many problems! 🤡
Now, let's get down to business. To convert this equation into standard form, we need to complete the square for both the x and y terms.
Starting with the x terms:
4x² - 20x
We can complete the square by adding and subtracting the square of half the coefficient of x (which is -20/2 = -10). So, we get:
4(x² - 5x + 25) - 100
Now, let's do the same for the y terms:
4y² + 12y
Adding and subtracting the square of half the coefficient of y (which is 12/2 = 6), we have:
4(y² + 6y + 9) - 36
Combining everything and simplifying, we get:
4(x² - 5x + 25) + 4(y² + 6y + 9) - 100 - 36 - 2 = 0
Simplifying further:
4(x² - 5x + 25 + y² + 6y + 9) - 138 =0
Finally, in standard form:
4(x² + y² - 5x + 6y + 34) - 138 = 0
There you have it! The standard form of the given equation is 4(x² + y² - 5x + 6y + 34) - 138 = 0.
The standard form of a quadratic equation in two variables, x and y, is given by:
Ax² + By² + Cx + Dy + E = 0
To rewrite the equation 4x²+4y²-20x+12y-2=0 in standard form, we need to group the x-terms together, the y-terms together, and move the constant term (in this case, -2) to the other side of the equation.
Let's start by rearranging the terms:
4x² - 20x + 4y² + 12y = 2
Now, we can rewrite this equation by completing the square for both the x-terms and y-terms separately. Completing the square involves adding a constant value to both sides of an equation to make a perfect square trinomial.
For the x-terms:
First, we factor out the coefficient of x², which is 4:
4(x² - 5x) + 4y² + 12y = 2
To complete the square for the x-terms, we take half of the coefficient of x, which is -5, square it, and add it to both sides of the equation:
4(x² - 5x + (-5/2)²) + 4y² + 12y = 2 + 4 * (-5/2)²
Simplifying this gives us:
4(x² - 5x + 25/4) + 4y² + 12y = 2 + 4 * 25/4
Next, we can rewrite the x-terms as a perfect square trinomial:
4(x - 5/2)² + 4y² + 12y = 2 + 25
Now, let's do the same process for the y-terms:
4(x - 5/2)² + 4(y² + 3y) = 27
For the y-terms, we take half of the coefficient of y, which is 3, square it, and add it to both sides of the equation:
4(x - 5/2)² + 4(y² + 3y + (3/2)²) = 27 + 4 * (3/2)²
Simplifying this gives us:
4(x - 5/2)² + 4(y² + 3y + 9/4) = 27 + 4 * 9/4
Now, we can rewrite the y-terms as a perfect square trinomial:
4(x - 5/2)² + 4(y + 3/2)² = 27 + 9
Rearranging the equation, we get:
4(x - 5/2)² + 4(y + 3/2)² = 36
Finally, we divide both sides of the equation by the constant term, 36, to isolate the equation on the left side:
(x - 5/2)² + (y + 3/2)² = 9
Therefore, the standard form of the equation 4x²+4y²-20x+12y-2=0 is (x - 5/2)² + (y + 3/2)² = 9.