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Sagot :
Certainly! Let's examine each equation to determine if it is quadratic.
An equation is quadratic if it can be written in the standard form [tex]\(ax^2 + bx + c = 0\)[/tex], where [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are constants, and [tex]\(a \neq 0\)[/tex].
### Equation 1: [tex]\(3x + 8 = 15\)[/tex]
Step-by-step:
1. Subtract 15 from both sides to get it in standard form: [tex]\(3x + 8 - 15 = 0\)[/tex].
2. Simplify: [tex]\(3x - 7 = 0\)[/tex].
This is a linear equation (degree 1), not a quadratic equation (degree 2).
Conclusion: This equation is not quadratic.
### Equation 2: [tex]\(x^2 - 5x + 10 = 0\)[/tex]
Step-by-step:
1. This equation is already in standard form [tex]\(ax^2 + bx + c = 0\)[/tex] with [tex]\(a = 1\)[/tex], [tex]\(b = -5\)[/tex], and [tex]\(c = 10\)[/tex].
Since [tex]\(a \neq 0\)[/tex] (specifically, [tex]\(a = 1\)[/tex]), this is indeed a quadratic equation.
Conclusion: This equation is quadratic.
### Equation 3: [tex]\(12 - 4x = 0\)[/tex]
Step-by-step:
1. We can rewrite it in standard form by moving all terms to one side: [tex]\(-4x + 12 = 0\)[/tex].
2. Alternatively, [tex]\(-4x + 12 = 0\)[/tex] can be expressed as [tex]\(12 - 4x = 0\)[/tex].
This is a linear equation (degree 1), not a quadratic equation.
Conclusion: This equation is not quadratic.
### Equation 4: [tex]\(2x^2 - 7x = 12\)[/tex]
Step-by-step:
1. Subtract 12 from both sides to get it in standard form: [tex]\(2x^2 - 7x - 12 = 0\)[/tex].
Now, we see that it is in the standard form [tex]\(ax^2 + bx + c = 0\)[/tex] with [tex]\(a = 2\)[/tex], [tex]\(b = -7\)[/tex], and [tex]\(c = -12\)[/tex].
Since [tex]\(a \neq 0\)[/tex] (specifically, [tex]\(a = 2\)[/tex]), this is a quadratic equation.
Conclusion: This equation is quadratic.
### Equation 5: [tex]\(6 - 2x + 3x^2 = 0\)[/tex]
Step-by-step:
1. This equation is already in standard form [tex]\(ax^2 + bx + c = 0\)[/tex] with [tex]\(a = 3\)[/tex], [tex]\(b = -2\)[/tex], and [tex]\(c = 6\)[/tex].
Since [tex]\(a \neq 0\)[/tex] (specifically, [tex]\(a = 3\)[/tex]), this is a quadratic equation.
Conclusion: This equation is quadratic.
### Summary
1. [tex]\(3x + 8 = 15\)[/tex] is not quadratic.
2. [tex]\(x^2 - 5x + 10 = 0\)[/tex] is quadratic.
3. [tex]\(12 - 4x = 0\)[/tex] is not quadratic.
4. [tex]\(2x^2 - 7x = 12\)[/tex] is quadratic.
5. [tex]\(6 - 2x + 3x^2 = 0\)[/tex] is quadratic.
An equation is quadratic if it can be written in the standard form [tex]\(ax^2 + bx + c = 0\)[/tex], where [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are constants, and [tex]\(a \neq 0\)[/tex].
### Equation 1: [tex]\(3x + 8 = 15\)[/tex]
Step-by-step:
1. Subtract 15 from both sides to get it in standard form: [tex]\(3x + 8 - 15 = 0\)[/tex].
2. Simplify: [tex]\(3x - 7 = 0\)[/tex].
This is a linear equation (degree 1), not a quadratic equation (degree 2).
Conclusion: This equation is not quadratic.
### Equation 2: [tex]\(x^2 - 5x + 10 = 0\)[/tex]
Step-by-step:
1. This equation is already in standard form [tex]\(ax^2 + bx + c = 0\)[/tex] with [tex]\(a = 1\)[/tex], [tex]\(b = -5\)[/tex], and [tex]\(c = 10\)[/tex].
Since [tex]\(a \neq 0\)[/tex] (specifically, [tex]\(a = 1\)[/tex]), this is indeed a quadratic equation.
Conclusion: This equation is quadratic.
### Equation 3: [tex]\(12 - 4x = 0\)[/tex]
Step-by-step:
1. We can rewrite it in standard form by moving all terms to one side: [tex]\(-4x + 12 = 0\)[/tex].
2. Alternatively, [tex]\(-4x + 12 = 0\)[/tex] can be expressed as [tex]\(12 - 4x = 0\)[/tex].
This is a linear equation (degree 1), not a quadratic equation.
Conclusion: This equation is not quadratic.
### Equation 4: [tex]\(2x^2 - 7x = 12\)[/tex]
Step-by-step:
1. Subtract 12 from both sides to get it in standard form: [tex]\(2x^2 - 7x - 12 = 0\)[/tex].
Now, we see that it is in the standard form [tex]\(ax^2 + bx + c = 0\)[/tex] with [tex]\(a = 2\)[/tex], [tex]\(b = -7\)[/tex], and [tex]\(c = -12\)[/tex].
Since [tex]\(a \neq 0\)[/tex] (specifically, [tex]\(a = 2\)[/tex]), this is a quadratic equation.
Conclusion: This equation is quadratic.
### Equation 5: [tex]\(6 - 2x + 3x^2 = 0\)[/tex]
Step-by-step:
1. This equation is already in standard form [tex]\(ax^2 + bx + c = 0\)[/tex] with [tex]\(a = 3\)[/tex], [tex]\(b = -2\)[/tex], and [tex]\(c = 6\)[/tex].
Since [tex]\(a \neq 0\)[/tex] (specifically, [tex]\(a = 3\)[/tex]), this is a quadratic equation.
Conclusion: This equation is quadratic.
### Summary
1. [tex]\(3x + 8 = 15\)[/tex] is not quadratic.
2. [tex]\(x^2 - 5x + 10 = 0\)[/tex] is quadratic.
3. [tex]\(12 - 4x = 0\)[/tex] is not quadratic.
4. [tex]\(2x^2 - 7x = 12\)[/tex] is quadratic.
5. [tex]\(6 - 2x + 3x^2 = 0\)[/tex] is quadratic.
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