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Part 1
Suppose the number of students in a mathematics class is x. The teacher insists that each student participate in group work each class. The number of possible groups​ is:
Upper G equals StartFraction x squared minus 5 x plus 7 Over 3 EndFractionG=
x2−5x+7
3
A teacher claims each student could be in 19 different groups. How many students are​ there?

We are given the formula for the number of possible groups \( G \):

\[
G = \frac{x^2 - 5x + 7}{3}
\]

where \( x \) is the number of students in the class. The teacher claims that each student could be in 19 different groups, meaning \( G = 19 \). We need to determine the value of \( x \) (the number of students).

### Step 1: Set up the equation
We equate the formula for \( G \) to 19:

\[
\frac{x^2 - 5x + 7}{3} = 19
\]

### Step 2: Eliminate the fraction
To eliminate the denominator, multiply both sides of the equation by 3:

\[
x^2 - 5x + 7 = 19 \times 3
\]

\[
x^2 - 5x + 7 = 57
\]

### Step 3: Rearrange the equation
Now subtract 57 from both sides to set the equation to zero:

\[
x^2 - 5x + 7 - 57 = 0
\]

\[
x^2 - 5x - 50 = 0
\]

### Step 4: Solve the quadratic equation
We now solve the quadratic equation \( x^2 - 5x - 50 = 0 \) using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

For the equation \( x^2 - 5x - 50 = 0 \), we have \( a = 1 \), \( b = -5 \), and \( c = -50 \).

### Step 5: Calculate the discriminant
First, calculate the discriminant \( \Delta = b^2 - 4ac \):

\[
\Delta = (-5)^2 - 4(1)(-50) = 25 + 200 = 225
\]

### Step 6: Find the roots
Now substitute the values into the quadratic formula:

\[
x = \frac{-(-5) \pm \sqrt{225}}{2(1)}
\]

\[
x = \frac{5 \pm 15}{2}
\]

So we have two possible solutions:

\[
x = \frac{5 + 15}{2} = \frac{20}{2} = 10
\]

\[
x = \frac{5 - 15}{2} = \frac{-10}{2} = -5
\]

Since the number of students \( x \) must be positive, we discard \( x = -5 \) and keep \( x = 10 \).

### Conclusion:
There are \( 10 \) students in the class.

Would you like further details or have any questions?

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Here are 5 related questions:

1. How can you derive a general formula for group formation?
2. What if each student could be in 30 groups—how would the result change?
3. What happens if the quadratic equation has no real solutions?
4. How do you interpret negative discriminants in quadratic equations?
5. Can you solve quadratic equations using factorization instead of the quadratic formula?

**Tip:** Always check the discriminant to determine the nature of the solutions (real or complex) before solving a quadratic equation.

Suppose the number of students in a mathematics class is x. The teacher insists that each student participate in group work each class. The number of possible groups​ is: G = (x^2 − 5x + 7) / 3. A teacher claims each student could be in 19 different groups. How many students are​ there?

Explore how to solve the quadratic equation (x^2 - 5x + 7) / 3 = 19 to find the number of students in a class. This solution uses the quadratic formula and covers key algebraic concepts suitable for students in Grades 9-11.

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