A system of quadratic equations is a set of two equations that each include a variable that is raised to the second power, like x^2 or y^2. These equations can be written in the form of:
ax^2 + by^2 = c dx + ey = f
To solve this kind of system, you'll need to use a method called substitution. Here's an example of how it works:
Let's say we have the following system of equations:
2x^2 + y^2 = 13 x + y = 5
First, we can use the second equation to solve for one of the variables in terms of the other. In this case, we can solve for x:
x = 5 - y
Next, we can substitute that expression for x into the first equation:
2(5-y)^2 + y^2 = 13
Now, we just need to solve for y. We can simplify the left side of the equation by expanding the square:
2(25 - 10y + y^2) + y^2 = 13 50 - 20y + 3y^2 = 13
This simplifies to:
3y^2 - 20y + 37 = 0
We can solve for y using the quadratic formula:
y = (20 ± √(-20^2 - 4337)) / (2*3)
After doing the math, we get two solutions for y:
y = 2 or y = (5/3)
Now we can plug each value of y back into the equation we used to solve for x:
If y = 2, then x = 3 If y = (5/3), then x = (10/3)
So the solutions to this system are:
x = 3, y = 2 x = (10/3), y = (5/3)
That's how you can solve a system of quadratic equations using substitution.