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The problem asks to find the values of \( a \), \( b \), and \( c \) in the quadratic equation:

\[
y = ax^2 + bx + c
\]

given the following conditions:
1. The graph has a minimum at the point \( (5, -3) \).
2. The graph passes through the point \( (4, 0) \).

### Step 1: Use the vertex form of a parabola
The minimum point \( (5, -3) \) tells us that the vertex of the parabola is at \( x = 5 \). The vertex form of a quadratic function is:

\[
y = a(x - h)^2 + k
\]

where \( (h, k) \) is the vertex. In this case, \( h = 5 \) and \( k = -3 \), so we can write the equation as:

\[
y = a(x - 5)^2 - 3
\]

### Step 2: Use the point \( (4, 0) \) to find \( a \)
Now, substitute \( x = 4 \) and \( y = 0 \) (the point \( (4, 0) \)) into the equation:

\[
0 = a(4 - 5)^2 - 3
\]
\[
0 = a(1)^2 - 3
\]
\[
a - 3 = 0
\]
\[
a = 3
\]

### Step 3: Write the equation in standard form
Now that we know \( a = 3 \), we can substitute this value back into the vertex form:

\[
y = 3(x - 5)^2 - 3
\]

Expand the equation:

\[
y = 3(x^2 - 10x + 25) - 3
\]
\[
y = 3x^2 - 30x + 75 - 3
\]
\[
y = 3x^2 - 30x + 72
\]

Thus, the quadratic equation is:

\[
y = 3x^2 - 30x + 72
\]

### Conclusion
The values of \( a \), \( b \), and \( c \) are:
- \( a = 3 \)
- \( b = -30 \)
- \( c = 72 \)

Would you like any further details or explanations?

### Related Questions:
1. How do we know the point \( (5, -3) \) represents a minimum for this parabola?
2. What other forms of quadratic equations are useful for solving vertex problems?
3. How can you verify if a point lies on the graph of a quadratic function?
4. What is the significance of the discriminant in quadratic equations?
5. How can we find the axis of symmetry of any quadratic function?

### Tip:
When given the vertex of a quadratic function, using the vertex form of the equation can make solving much simpler!

The graph of y = ax^2 + bx + c has a minimum at (5, -3) and passes through (4, 0). Find the values of a, b, and c.

Solve the quadratic equation y = ax^2 + bx + c with a minimum point at (5, -3) and passing through (4, 0). Learn how to determine the values of a, b, and c using algebraic methods and vertex form.

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