To analyze the quadratic function \( f(x) = 3x^2 + 12x - 21 \), we'll proceed step-by-step to find the number of distinct real roots, the vertex, and the x-intercepts.
Step 1: Predict the number of distinct real roots.
The number of distinct real roots of a quadratic function can be determined using the discriminant \( D \), calculated as follows:
\[ D = b^2 - 4ac \]
For the function \( f(x) = 3x^2 + 12x - 21 \):
- \( a = 3 \)
- \( b = 12 \)
- \( c = -21 \)
Calculating the discriminant:
\[ D = 12^2 - 4(3)(-21) = 144 + 252 = 396 \]
Since \( D > 0 \), this indicates that there are two distinct real roots.
Step 2: Find the vertex.
The vertex \( (h, k) \) of a quadratic function in the form \( f(x) = ax^2 + bx + c \) can be found using the formula for \( h \):
\[ h = -\frac{b}{2a} \]
Substituting the values for \( a \) and \( b \):
\[ h = -\frac{12}{2 \times 3} = -\frac{12}{6} = -2 \]
Next, we find \( k \) by substituting \( h \) back into the function:
\[ k = f(-2) = 3(-2)^2 + 12(-2) - 21 \]
Calculating:
\[ k = 3(4) - 24 - 21 = 12 - 24 - 21 = -33 \]
Thus, the vertex of the parabola is at \( (-2, -33) \).
Step 3: Find the x-intercepts.
To find the x-intercepts, we set \( f(x) = 0 \) and solve for \( x \):
\[ 3x^2 + 12x - 21 = 0 \]
Dividing the entire equation by 3 gives:
\[ x^2 + 4x - 7 = 0 \]
Now using the quadratic formula \( x = \frac{-b \pm \sqrt{D}}{2a} \):
Here, \( a = 1 \), \( b = 4 \), and the discriminant \( D = 396 \).
Calculating the x-intercepts:
\[ x = \frac{-4 \pm \sqrt{396}}{2(1)} = \frac{-4 \pm \sqrt{396}}{2} \]
Simplifying \( \sqrt{396} \):
\[ \sqrt{396} = \sqrt{4 \times 99} = 2\sqrt{99} \]
Thus:
\[ x = \frac{-4 \pm 2\sqrt{99}}{2} = -2 \pm \sqrt{99} \]
Calculating the approximate values:
\[ \sqrt{99} \approx 9.95 \quad (\text{approximately}) \]
So the x-intercepts are:
\[ x \approx -2 + 9.95 \quad \text{and} \quad x \approx -2 - 9.95 \] \[ x \approx 7.95 \quad \text{and} \quad x \approx -11.95 \]
Summary of Results:
- Number of distinct real roots: 2
- Vertex: (-2, -33)
- X-intercepts: \( x \approx 7.95 \) and \( x \approx -11.95 \)
Now you can confirm these results using a graphing calculator and revise them if necessary!