To determine which of the given equations are quadratic equations, we need to analyze their forms and verify if they meet the criteria for being quadratic equations. A quadratic equation is a polynomial equation of the second degree, which means it must be of the form [tex]\(ax^2 + bx + c = 0\)[/tex], where [tex]\(a \neq 0\)[/tex].
Let's look at each option one by one.
### Option (a): [tex]\(x^3 - x = x^2 + 2\)[/tex]
To begin, we'll rearrange the terms to bring all terms to one side of the equation.
[tex]\[ x^3 - x - x^2 - 2 = 0 \][/tex]
Next, we combine like terms.
[tex]\[ x^3 - x^2 - x - 2 = 0 \][/tex]
The resulting equation is [tex]\(x^3 - x^2 - x - 2 = 0\)[/tex]. This equation is a polynomial equation of degree 3 (cubic), as the highest power of [tex]\(x\)[/tex] is 3. A quadratic equation must have a highest power of [tex]\(x\)[/tex] equal to 2.
Therefore, [tex]\(x^3 - x = x^2 + 2\)[/tex] is not a quadratic equation.
### Option (b): [tex]\(\sqrt{x + 4} = x + 1\)[/tex]
First, let's eliminate the square root by squaring both sides of the equation.
[tex]\[ (\sqrt{x + 4})^2 = (x + 1)^2 \][/tex]
[tex]\[ x + 4 = x^2 + 2x + 1 \][/tex]
Next, rearrange the terms to bring all of them to one side of the equation.
[tex]\[ x + 4 - x^2 - 2x - 1 = 0 \][/tex]
Then, combine like terms.
[tex]\[ -x^2 - x + 3 = 0 \][/tex]
Or equivalently,
[tex]\[ x^2 + x - 3 = 0 \][/tex]
This equation is a polynomial equation of degree 2, as the highest power of [tex]\(x\)[/tex] is 2. It fits the form [tex]\(ax^2 + bx + c = 0\)[/tex].
Therefore, [tex]\(\sqrt{x + 4} = x + 1\)[/tex] is a quadratic equation.
### Conclusion:
- Option (a) [tex]\(x^3 - x = x^2 + 2\)[/tex] is not a quadratic equation.
- Option (b) [tex]\(\sqrt{x + 4} = x + 1\)[/tex] is a quadratic equation.
