Answer :
Sure, let's break each of the equations down step-by-step to determine whether they are quadratic equations. A quadratic equation is generally of the form [tex]\(ax^2 + bx + c = 0\)[/tex], where [tex]\(a \neq 0\)[/tex].
1. Equation: [tex]\((2x - 5) + 2\)[/tex]
- This simplifies to [tex]\(2x - 3 = 0\)[/tex].
- This is a linear equation because it does not have an [tex]\(x^2\)[/tex] term.
- Not Quadratic [tex]\( \square \)[/tex]
2. Equation: [tex]\(3x^2 - 10 = 4x\)[/tex]
- Rearrange to the standard form: [tex]\(3x^2 - 4x - 10 = 0\)[/tex].
- This is a quadratic equation because it has an [tex]\(x^2\)[/tex] term.
- Quadratic [tex]\( \checkmark \)[/tex]
3. Equation: [tex]\((x - 5)^2 = 2\)[/tex]
- Expand and simplify: [tex]\(x^2 - 10x + 25 = 2\)[/tex], which becomes [tex]\(x^2 - 10x + 23 = 0\)[/tex].
- This is a quadratic equation because it has an [tex]\(x^2\)[/tex] term.
- Quadratic [tex]\( \checkmark \)[/tex]
4. Equation: [tex]\(5x(x - 7) = 0\)[/tex]
- Distribute: [tex]\(5x^2 - 35x = 0\)[/tex].
- This is a quadratic equation because it has an [tex]\(x^2\)[/tex] term.
- Quadratic [tex]\( \checkmark \)[/tex]
5. Equation: [tex]\(x(3x^2 - 2x + 1) = 0\)[/tex]
- Distribute: [tex]\(3x^3 - 2x^2 + x = 0\)[/tex].
- This is a cubic equation because it has an [tex]\(x^3\)[/tex] term.
- Not Quadratic [tex]\( \square \)[/tex]
6. Equation: [tex]\(-8 = (3x + 7)^2\)[/tex]
- Rearrange to [tex]\((3x + 7)^2 + 8 = 0\)[/tex].
- Expand: [tex]\(9x^2 + 42x + 49 + 8 = 0\)[/tex], which simplifies to [tex]\(9x^2 + 42x + 57 = 0\)[/tex].
- This is a quadratic equation because it has an [tex]\(x^2\)[/tex] term.
- Quadratic [tex]\( \checkmark \)[/tex]
7. Equation: [tex]\(15 + 4x^2 - 2x = 0\)[/tex]
- This simplifies to the standard form: [tex]\(4x^2 - 2x + 15 = 0\)[/tex].
- This is a quadratic equation because it has an [tex]\(x^2\)[/tex] term.
- Quadratic [tex]\( \checkmark \)[/tex]
8. Equation: [tex]\(82(4 - x^2)^2 = 3x\)[/tex]
- This would require expanding and rearranging but generally, it is not in the standard quadratic form.
- This is not a quadratic equation because it results in a higher degree polynomial.
- Not Quadratic [tex]\( \square \)[/tex]
9. Equation: [tex]\(5(3 - 2x) = 5x\)[/tex]
- Distribute: [tex]\(15 - 10x = 5x\)[/tex].
- Simplify and rearrange: [tex]\(15 = 15x\)[/tex], which is linear.
- This is a linear equation because it does not have an [tex]\(x^2\)[/tex] term.
- Not Quadratic [tex]\( \square \)[/tex]
10. Equation: [tex]\(2\left|(4x - 1)^2\right| = 8\)[/tex]
- Simplify: [tex]\(|(4x - 1)^2| = 4\)[/tex].
- The absolute value does not contribute to it being quadratic by structure.
- Generally, this does not conform to the standard quadratic equation format.
- Not Quadratic [tex]\( \square \)[/tex]
So, our final answers would be:
1. [tex]\( \square \)[/tex]
2. [tex]\( \checkmark \)[/tex]
3. [tex]\( \checkmark \)[/tex]
4. [tex]\( \checkmark \)[/tex]
5. [tex]\( \square \)[/tex]
6. [tex]\( \checkmark \)[/tex]
7. [tex]\( \checkmark \)[/tex]
8. [tex]\( \square \)[/tex]
9. [tex]\( \square \)[/tex]
10. [tex]\( \square \)[/tex]
1. Equation: [tex]\((2x - 5) + 2\)[/tex]
- This simplifies to [tex]\(2x - 3 = 0\)[/tex].
- This is a linear equation because it does not have an [tex]\(x^2\)[/tex] term.
- Not Quadratic [tex]\( \square \)[/tex]
2. Equation: [tex]\(3x^2 - 10 = 4x\)[/tex]
- Rearrange to the standard form: [tex]\(3x^2 - 4x - 10 = 0\)[/tex].
- This is a quadratic equation because it has an [tex]\(x^2\)[/tex] term.
- Quadratic [tex]\( \checkmark \)[/tex]
3. Equation: [tex]\((x - 5)^2 = 2\)[/tex]
- Expand and simplify: [tex]\(x^2 - 10x + 25 = 2\)[/tex], which becomes [tex]\(x^2 - 10x + 23 = 0\)[/tex].
- This is a quadratic equation because it has an [tex]\(x^2\)[/tex] term.
- Quadratic [tex]\( \checkmark \)[/tex]
4. Equation: [tex]\(5x(x - 7) = 0\)[/tex]
- Distribute: [tex]\(5x^2 - 35x = 0\)[/tex].
- This is a quadratic equation because it has an [tex]\(x^2\)[/tex] term.
- Quadratic [tex]\( \checkmark \)[/tex]
5. Equation: [tex]\(x(3x^2 - 2x + 1) = 0\)[/tex]
- Distribute: [tex]\(3x^3 - 2x^2 + x = 0\)[/tex].
- This is a cubic equation because it has an [tex]\(x^3\)[/tex] term.
- Not Quadratic [tex]\( \square \)[/tex]
6. Equation: [tex]\(-8 = (3x + 7)^2\)[/tex]
- Rearrange to [tex]\((3x + 7)^2 + 8 = 0\)[/tex].
- Expand: [tex]\(9x^2 + 42x + 49 + 8 = 0\)[/tex], which simplifies to [tex]\(9x^2 + 42x + 57 = 0\)[/tex].
- This is a quadratic equation because it has an [tex]\(x^2\)[/tex] term.
- Quadratic [tex]\( \checkmark \)[/tex]
7. Equation: [tex]\(15 + 4x^2 - 2x = 0\)[/tex]
- This simplifies to the standard form: [tex]\(4x^2 - 2x + 15 = 0\)[/tex].
- This is a quadratic equation because it has an [tex]\(x^2\)[/tex] term.
- Quadratic [tex]\( \checkmark \)[/tex]
8. Equation: [tex]\(82(4 - x^2)^2 = 3x\)[/tex]
- This would require expanding and rearranging but generally, it is not in the standard quadratic form.
- This is not a quadratic equation because it results in a higher degree polynomial.
- Not Quadratic [tex]\( \square \)[/tex]
9. Equation: [tex]\(5(3 - 2x) = 5x\)[/tex]
- Distribute: [tex]\(15 - 10x = 5x\)[/tex].
- Simplify and rearrange: [tex]\(15 = 15x\)[/tex], which is linear.
- This is a linear equation because it does not have an [tex]\(x^2\)[/tex] term.
- Not Quadratic [tex]\( \square \)[/tex]
10. Equation: [tex]\(2\left|(4x - 1)^2\right| = 8\)[/tex]
- Simplify: [tex]\(|(4x - 1)^2| = 4\)[/tex].
- The absolute value does not contribute to it being quadratic by structure.
- Generally, this does not conform to the standard quadratic equation format.
- Not Quadratic [tex]\( \square \)[/tex]
So, our final answers would be:
1. [tex]\( \square \)[/tex]
2. [tex]\( \checkmark \)[/tex]
3. [tex]\( \checkmark \)[/tex]
4. [tex]\( \checkmark \)[/tex]
5. [tex]\( \square \)[/tex]
6. [tex]\( \checkmark \)[/tex]
7. [tex]\( \checkmark \)[/tex]
8. [tex]\( \square \)[/tex]
9. [tex]\( \square \)[/tex]
10. [tex]\( \square \)[/tex]
