Answer :
Let's solve each of the given equations to find their roots and determine which equation has the roots 0 and 3.
### Equation \( A: (x+1)(x-3)=0 \)
To find the roots of this equation, we set each factor equal to zero:
[tex]\[ (x + 1) = 0 \][/tex]
[tex]\[ x = -1 \][/tex]
[tex]\[ (x - 3) = 0 \][/tex]
[tex]\[ x = 3 \][/tex]
The roots are \(-1\) and \(3\).
### Equation \( B: x^2 - 9 = 0 \)
To solve this equation, we can factor it or use the difference of squares:
[tex]\[ x^2 - 9 = (x - 3)(x + 3) = 0 \][/tex]
Setting each factor to zero:
[tex]\[ (x - 3) = 0 \][/tex]
[tex]\[ x = 3 \][/tex]
[tex]\[ (x + 3) = 0 \][/tex]
[tex]\[ x = -3 \][/tex]
The roots are \(3\) and \(-3\).
### Equation \( C: 3x(x-3)=0 \)
To find the roots, we set each factor equal to zero:
[tex]\[ 3x = 0 \][/tex]
[tex]\[ x = 0 \][/tex]
[tex]\[ (x - 3) = 0 \][/tex]
[tex]\[ x = 3 \][/tex]
The roots are \(0\) and \(3\).
### Equation \( D: (x+1)(x+3)=0 \)
To find the roots, we set each factor equal to zero:
[tex]\[ (x + 1) = 0 \][/tex]
[tex]\[ x = -1 \][/tex]
[tex]\[ (x + 3) = 0 \][/tex]
[tex]\[ x = -3 \][/tex]
The roots are \(-1\) and \(-3\).
### Conclusion
From the roots obtained for each equation:
- Equation \( A \) has roots \(-1\) and \(3\).
- Equation \( B \) has roots \(3\) and \(-3\).
- Equation \( C \) has roots \(0\) and \(3\).
- Equation \( D \) has roots \(-1\) and \(-3\).
The equation with roots [tex]\(0\)[/tex] and [tex]\(3\)[/tex] is Equation [tex]\( C: 3x(x-3)=0 \)[/tex].
### Equation \( A: (x+1)(x-3)=0 \)
To find the roots of this equation, we set each factor equal to zero:
[tex]\[ (x + 1) = 0 \][/tex]
[tex]\[ x = -1 \][/tex]
[tex]\[ (x - 3) = 0 \][/tex]
[tex]\[ x = 3 \][/tex]
The roots are \(-1\) and \(3\).
### Equation \( B: x^2 - 9 = 0 \)
To solve this equation, we can factor it or use the difference of squares:
[tex]\[ x^2 - 9 = (x - 3)(x + 3) = 0 \][/tex]
Setting each factor to zero:
[tex]\[ (x - 3) = 0 \][/tex]
[tex]\[ x = 3 \][/tex]
[tex]\[ (x + 3) = 0 \][/tex]
[tex]\[ x = -3 \][/tex]
The roots are \(3\) and \(-3\).
### Equation \( C: 3x(x-3)=0 \)
To find the roots, we set each factor equal to zero:
[tex]\[ 3x = 0 \][/tex]
[tex]\[ x = 0 \][/tex]
[tex]\[ (x - 3) = 0 \][/tex]
[tex]\[ x = 3 \][/tex]
The roots are \(0\) and \(3\).
### Equation \( D: (x+1)(x+3)=0 \)
To find the roots, we set each factor equal to zero:
[tex]\[ (x + 1) = 0 \][/tex]
[tex]\[ x = -1 \][/tex]
[tex]\[ (x + 3) = 0 \][/tex]
[tex]\[ x = -3 \][/tex]
The roots are \(-1\) and \(-3\).
### Conclusion
From the roots obtained for each equation:
- Equation \( A \) has roots \(-1\) and \(3\).
- Equation \( B \) has roots \(3\) and \(-3\).
- Equation \( C \) has roots \(0\) and \(3\).
- Equation \( D \) has roots \(-1\) and \(-3\).
The equation with roots [tex]\(0\)[/tex] and [tex]\(3\)[/tex] is Equation [tex]\( C: 3x(x-3)=0 \)[/tex].
