To find the quadratic equation $ax^2 + bx + c$ that passes through the points $(0, 10)$, $(4, 10.5)$, and $(12, 8)$, we can substitute these points into the general form of the quadratic equation to create a system of equations.

Given:

1. $(0, 10)$
2. $(4, 10.5)$
3. $(12, 8)$

Step 1: Substitute the Points into the Equation

For $(0, 10)$: $a(0)^2 + b(0) + c = 10 \implies c = 10$

For $(4, 10.5)$: $a(4)^2 + b(4) + c = 10.5 \implies 16a + 4b + c = 10.5$

For $(12, 8)$: $a(12)^2 + b(12) + c = 8 \implies 144a + 12b + c = 8$

Step 2: Substitute $c = 10$ into the Equations

1. $16a + 4b + 10 = 10.5 \implies 16a + 4b = 0.5$
2. $144a + 12b + 10 = 8 \implies 144a + 12b = -2$

Step 3: Solve the System of Equations

First equation: $16a + 4b = 0.5 \quad \text{(Equation 1)}$

Second equation: $144a + 12b = -2 \quad \text{(Equation 2)}$

To solve these equations simultaneously, we can first simplify Equation 1 by dividing it by 4: $4a + b = 0.125 \quad \text{(Equation 3)}$

Similarly, simplify Equation 2 by dividing it by 12: $12a + b = -\frac{1}{6} \quad \text{(Equation 4)}$

Now, subtract Equation 3 from Equation 4: $(12a + b) - (4a + b) = -\frac{1}{6} - 0.125$ $8a = -\frac{1}{6} - \frac{1}{8}$

Find a common denominator: $8a = -\frac{4}{24} - \frac{3}{24} = -\frac{7}{24}$

$a = -\frac{7}{24 \times 8} = -\frac{7}{192}$

Step 4: Substitute $a = -\frac{7}{192}$ back into Equation 3

$4\left(-\frac{7}{192}\right) + b = 0.125$ $-\frac{28}{192} + b = 0.125$ $b = 0.125 + \frac{28}{192}$ $b = \frac{24}{192} + \frac{28}{192} = \frac{52}{192} = \frac{13}{48}$

Final Equation

Now substitute the values of $a$, $b$, and $c$ into the original quadratic equation: $y = -\frac{7}{192}x^2 + \frac{13}{48}x + 10$

This is the quadratic equation that passes through the given points.

Would you like a detailed step-by-step verification or have any questions?

Here are 8 related questions:

1. How can we check the accuracy of this quadratic equation?
2. What are the x-intercepts of this quadratic equation?
3. How do we determine the vertex of this parabola?
4. Can this equation be expressed in vertex form?
5. How does changing the points affect the coefficients?
6. How would the solution differ if the points were collinear?
7. What is the significance of each coefficient in the quadratic equation?
8. How can this method be generalized for any three points?

Tip: Always check the discriminant when solving quadratic equations to determine the nature of the roots (real or complex).