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 are the values of x for which the quadratic function is equal to zero. These roots can also be referred to as the x-intercepts or zeros of the corresponding function.
Let's consider the quadratic equation:
f(x) = ax^2 + bx + c
To find the roots of this equation, we set f(x) equal to zero and solve for x:
ax^2 + bx + c = 0
For example, let's take the quadratic equation:
f(x) = x^2 - 5x + 6
To find the roots, we set f(x) equal to zero:
x^2 - 5x + 6 = 0
Now we need to solve for x. Factoring, we have:
(x - 2)(x - 3) = 0
This equation is satisfied when either (x - 2) = 0 or (x - 3) = 0. Solving each equation, we find that the roots are x = 2 and x = 3.
Therefore, the x-intercepts or zeros of the quadratic function f(x) = x^2 - 5x + 6 are x = 2 and x = 3.
In summary, the roots of a quadratic equation represent the x-values where the quadratic function equals zero. These roots are equivalent to the x-intercepts or zeros of the corresponding function.