To rewrite the quadratic equation fx = x² + 3x + 2 into vertex form, we will complete the square.
The vertex form of a quadratic equation is given by:
f(x) = a(x – h)² + k
where (h, k) is the vertex of the parabola. Let’s start with the given equation:
f(x) = x² + 3x + 2
1. First, we can group the x terms:
f(x) = (x² + 3x) + 2
2. Next, we complete the square for the expression x² + 3x. To do this, we take half of the coefficient of x, square it, and add and subtract that value inside the parentheses:
- The coefficient of x is 3. Half of 3 is 1.5, and squaring that gives us (1.5)² = 2.25.
3. Now we can rewrite the equation by adding and subtracting 2.25:
f(x) = (x² + 3x + 2.25 – 2.25) + 2
4. This simplifies to:
f(x) = ((x + 1.5)² – 2.25) + 2
5. Finally, we simplify it further:
f(x) = (x + 1.5)² – 0.25
Thus, the vertex form of the quadratic equation f(x) = x² + 3x + 2 is:
f(x) = (x + 1.5)² – 0.25
In this form, you can see the vertex of the parabola is at the point (-1.5, -0.25).