Answer :
To identify the quadratic equations from the given list, we need to recall that a quadratic equation is a polynomial of degree 2, which means it follows the general form:
[tex]\[ y = ax^2 + bx + c \][/tex]
where [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are constants, and [tex]\( a \neq 0 \)[/tex]. Let's analyze each given equation one by one to see if it matches this form.
1. Equation: [tex]\( y = 2x + 1 \)[/tex]
- This equation is a first-degree polynomial (linear equation) since the highest power of [tex]\( x \)[/tex] is [tex]\( x^1 \)[/tex].
- It does not have an [tex]\( x^2 \)[/tex] term.
- Therefore, this is not a quadratic equation.
2. Equation: [tex]\( y = 5x^2 + 3x - 1 \)[/tex]
- This equation includes an [tex]\( x^2 \)[/tex] term with a coefficient of 5.
- It also has an [tex]\( x \)[/tex] term with a coefficient of 3 and a constant term of -1.
- The general form [tex]\( y = ax^2 + bx + c \)[/tex] is clearly evident here with [tex]\( a = 5 \)[/tex], [tex]\( b = 3 \)[/tex], and [tex]\( c = -1 \)[/tex].
- Hence, this is a quadratic equation.
3. Equation: [tex]\( y = -\frac{1}{2}x^2 \)[/tex]
- This equation includes an [tex]\( x^2 \)[/tex] term with a coefficient of [tex]\( -\frac{1}{2} \)[/tex].
- It does not have an [tex]\( x \)[/tex] term or a constant term, but those are not required for it to be quadratic.
- The general form [tex]\( y = ax^2 + bx + c \)[/tex] is evident here with [tex]\( a = -\frac{1}{2} \)[/tex], [tex]\( b = 0 \)[/tex], and [tex]\( c = 0 \)[/tex].
- Hence, this is a quadratic equation.
4. Equation: [tex]\( y = \sqrt{x + 5} \)[/tex]
- This equation involves a square root, [tex]\( \sqrt{x + 5} \)[/tex], which means it is not represented in the polynomial form [tex]\( y = ax^2 + bx + c \)[/tex].
- The presence of a square root indicates that it is not a polynomial, let alone a quadratic polynomial.
- Therefore, this is not a quadratic equation.
Based on the analysis, the quadratic equations from the list are:
[tex]\[ y = 5x^2 + 3x - 1 \][/tex]
[tex]\[ y = -\frac{1}{2}x^2 \][/tex]
Thus, the quadratic equations identified are:
[tex]\[ \boxed{y = 5x^2 + 3x - 1 \quad \text{and} \quad y = -\frac{1}{2}x^2} \][/tex]
[tex]\[ y = ax^2 + bx + c \][/tex]
where [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are constants, and [tex]\( a \neq 0 \)[/tex]. Let's analyze each given equation one by one to see if it matches this form.
1. Equation: [tex]\( y = 2x + 1 \)[/tex]
- This equation is a first-degree polynomial (linear equation) since the highest power of [tex]\( x \)[/tex] is [tex]\( x^1 \)[/tex].
- It does not have an [tex]\( x^2 \)[/tex] term.
- Therefore, this is not a quadratic equation.
2. Equation: [tex]\( y = 5x^2 + 3x - 1 \)[/tex]
- This equation includes an [tex]\( x^2 \)[/tex] term with a coefficient of 5.
- It also has an [tex]\( x \)[/tex] term with a coefficient of 3 and a constant term of -1.
- The general form [tex]\( y = ax^2 + bx + c \)[/tex] is clearly evident here with [tex]\( a = 5 \)[/tex], [tex]\( b = 3 \)[/tex], and [tex]\( c = -1 \)[/tex].
- Hence, this is a quadratic equation.
3. Equation: [tex]\( y = -\frac{1}{2}x^2 \)[/tex]
- This equation includes an [tex]\( x^2 \)[/tex] term with a coefficient of [tex]\( -\frac{1}{2} \)[/tex].
- It does not have an [tex]\( x \)[/tex] term or a constant term, but those are not required for it to be quadratic.
- The general form [tex]\( y = ax^2 + bx + c \)[/tex] is evident here with [tex]\( a = -\frac{1}{2} \)[/tex], [tex]\( b = 0 \)[/tex], and [tex]\( c = 0 \)[/tex].
- Hence, this is a quadratic equation.
4. Equation: [tex]\( y = \sqrt{x + 5} \)[/tex]
- This equation involves a square root, [tex]\( \sqrt{x + 5} \)[/tex], which means it is not represented in the polynomial form [tex]\( y = ax^2 + bx + c \)[/tex].
- The presence of a square root indicates that it is not a polynomial, let alone a quadratic polynomial.
- Therefore, this is not a quadratic equation.
Based on the analysis, the quadratic equations from the list are:
[tex]\[ y = 5x^2 + 3x - 1 \][/tex]
[tex]\[ y = -\frac{1}{2}x^2 \][/tex]
Thus, the quadratic equations identified are:
[tex]\[ \boxed{y = 5x^2 + 3x - 1 \quad \text{and} \quad y = -\frac{1}{2}x^2} \][/tex]
