Question
Check Answer Question 15 Find \( b \) and \( c \) so that \( y=3 x^{2}+b x+c \) has vertex \( (-6,0) \). \( b=\square \) \( c=\square \)
Ask by Nunez Wilson. in the United States
Mar 10,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( b = 36 \) and \( c = 108 \).
Solution
1. Write the quadratic in vertex form:
\[
y = a(x-h)^2 + k,
\]
where \((h, k)\) is the vertex and \(a = 3\). Since the vertex is \((-6, 0)\), we have:
\[
y = 3(x - (-6))^2 + 0 = 3(x + 6)^2.
\]
2. Expand the expression:
\[
3(x + 6)^2 = 3(x^2 + 12x + 36) = 3x^2 + 36x + 108.
\]
3. Compare with the standard form:
\[
y = 3x^2 + bx + c.
\]
Equate the coefficients of like terms:
\[
b = 36 \quad \text{and} \quad c = 108.
\]
The answers are:
\[
b = 36, \quad c = 108.
\]
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Simplify this solution The Deep Dive
To find \( b \) and \( c \) for the curve \( y = 3x^2 + bx + c \) to have the vertex at \( (-6,0) \), we can use the vertex formula for a parabola in the form \( y = ax^2 + bx + c \). The x-coordinate of the vertex is given by \( x = -\frac{b}{2a} \). Here, \( a = 3 \). Substituting the vertex point: \[ -6 = -\frac{b}{2(3)} \implies -6 = -\frac{b}{6} \implies b = 36. \] Now, to find \( c \), we substitute the vertex point \( (-6, 0) \) back into the equation: \[ 0 = 3(-6)^2 + 36(-6) + c. \] \[ 0 = 3(36) - 216 + c \implies 0 = 108 - 216 + c \implies c = 108. \] Thus, the values are: \( b=36 \) \( c=108 \)
