Answer :
To determine if the given equation \( x^3 - x = x^2 + 2 \) is a quadratic equation, we need to rewrite it in standard polynomial form and analyze its degree.
1. Start with the given equation:
[tex]\[ x^3 - x = x^2 + 2 \][/tex]
2. Move all terms to one side of the equation to set it to zero:
[tex]\[ x^3 - x - x^2 - 2 = 0 \][/tex]
3. Rearrange the equation in terms of decreasing powers of \( x \):
[tex]\[ x^3 - x^2 - x - 2 = 0 \][/tex]
4. Identify the highest degree term. In the equation \( x^3 - x^2 - x - 2 = 0 \), the highest degree term is \( x^3 \), which indicates that this is a cubic equation because the highest power of \( x \) is 3.
To be quadratic, an equation must be of the form \( ax^2 + bx + c = 0 \) where the highest power of \( x \) is 2.
Since the degree of the polynomial is 3, the equation [tex]\( x^3 - x = x^2 + 2 \)[/tex] is not quadratic; it is a cubic equation.
1. Start with the given equation:
[tex]\[ x^3 - x = x^2 + 2 \][/tex]
2. Move all terms to one side of the equation to set it to zero:
[tex]\[ x^3 - x - x^2 - 2 = 0 \][/tex]
3. Rearrange the equation in terms of decreasing powers of \( x \):
[tex]\[ x^3 - x^2 - x - 2 = 0 \][/tex]
4. Identify the highest degree term. In the equation \( x^3 - x^2 - x - 2 = 0 \), the highest degree term is \( x^3 \), which indicates that this is a cubic equation because the highest power of \( x \) is 3.
To be quadratic, an equation must be of the form \( ax^2 + bx + c = 0 \) where the highest power of \( x \) is 2.
Since the degree of the polynomial is 3, the equation [tex]\( x^3 - x = x^2 + 2 \)[/tex] is not quadratic; it is a cubic equation.
