Hey, I just have two questions, Thanks.
1. Find the exact solutions of x² - (y - 6)² = 36 and y = -x²
2. Solve the system,4x² + y² = 20 and y = 4x.
#1 since you know that y = -x^2, just plug that in:
x^2 - (-x^2-6)^2 = 36
This has no real solutions. If you graph the hyperbola and the parabola, you will see that they do not intersect.
http://www.wolframalpha.com/input/?i=plot+x%C2%B2+-+%28y+-+6%29%C2%B2+%3D+36+%2C+y+%3D+-x%C2%B2++where-10+%3C%3D+x+%3C%3D+10
#2 y=4x, so
4x² + (4x)² = 20
20x^2 = 20
x = ±1
http://www.wolframalpha.com/input/?i=plot+4x%C2%B2+%2B+y%C2%B2+%3D+20+%2C+y+%3D+4x
Thank you so much, that was very helpful:)
1. To find the exact solutions of the equation x² - (y - 6)² = 36 and y = -x², we need to substitute the value of y from the second equation into the first equation.
Substituting y = -x² into the first equation:
x² - (-x² - 6)² = 36
Simplifying the expression inside the parentheses:
x² - (x⁴ + 12x² + 36) = 36
Rearranging the equation:
x⁴ + 13x² + 72 = 0
This is a fourth-degree polynomial equation, and solving it exactly requires advanced techniques such as factoring, the rational root theorem, or using a numerical method like the Newton-Raphson method. If you would like to solve it using one of these methods, please let me know.
2. In order to solve the system of equations 4x² + y² = 20 and y = 4x, we need to substitute the value of y from the second equation into the first equation.
Substituting y = 4x into the first equation:
4x² + (4x)² = 20
Simplifying the equation:
4x² + 16x² = 20
Combining like terms:
20x² = 20
Dividing both sides by 20:
x² = 1
Taking the square root of both sides:
x = ±1
Now that we have found the value of x, we can substitute it back into the second equation to find the corresponding values of y.
Substituting x = 1 into y = 4x:
y = 4(1)
y = 4
So one solution to the system of equations is x = 1, y = 4.
Substituting x = -1 into y = 4x:
y = 4(-1)
y = -4
So the other solution to the system of equations is x = -1, y = -4.
Therefore, the solutions to the system of equations are (1, 4) and (-1, -4).