Explain, using an example, how the roots of a quadratic equation are related to the x-intercepts and zeros of the corresponding function.
The roots of a quadratic equation and the x-intercepts or zeros of the corresponding function are essentially the same thing.
To understand this relationship, let's consider the quadratic equation:
y = x^2 - 4x + 3
To find the roots of this quadratic equation, we set y equal to zero and solve for x:
0 = x^2 - 4x + 3
Factoring this equation, we get:
0 = (x - 3)(x - 1)
Setting each factor equal to zero, we find the roots:
x - 3 = 0 ---> x = 3
x - 1 = 0 ---> x = 1
These two values, 3 and 1, are the roots of the quadratic equation.
Now, let's find the x-intercepts or zeros of the corresponding function y = x^2 - 4x + 3.
To do this, we set y equal to zero and solve for x:
0 = x^2 - 4x + 3
This equation is the same as the one we solved earlier for finding the roots. So the x-intercepts or zeros of the function are the same as the roots of the quadratic equation.
In this case, the x-intercepts or zeros of the function are x = 3 and x = 1, which are also the roots of the quadratic equation.
Therefore, the roots of a quadratic equation are directly related to the x-intercepts or zeros of the corresponding function.