To determine which of the given equations is not a quadratic equation, we first need to understand what defines a quadratic equation. A quadratic equation is a polynomial equation of degree 2, which means it generally takes the form:
[tex]\[ ax^2 + bx + c = 0 \][/tex]
where [tex]\(a \neq 0\)[/tex].
We will simplify each of the given equations and identify whether each simplified form retains a degree of 2.
### Option a) [tex]\( x^2 + 4x = 11 + x^2 \)[/tex]
1. Subtract [tex]\(x^2\)[/tex] from both sides:
[tex]\[ x^2 + 4x - x^2 = 11 + x^2 - x^2 \][/tex]
2. Simplify the equation:
[tex]\[ 4x = 11 \][/tex]
This equation is linear, not quadratic, as it can be written in the form [tex]\(4x = 11\)[/tex], which is a degree 1 equation.
### Option b) [tex]\( x^2 = 4x \)[/tex]
1. Bring all terms to one side to set the equation to zero:
[tex]\[ x^2 - 4x = 0 \][/tex]
This is a quadratic equation since it is in the form [tex]\(ax^2 + bx + c = 0\)[/tex] with [tex]\(a = 1\)[/tex], [tex]\(b = -4\)[/tex], and [tex]\(c = 0\)[/tex].
### Option c) [tex]\( 5x^2 = 90 \)[/tex]
1. Bring all terms to one side to set the equation to zero:
[tex]\[ 5x^2 - 90 = 0 \][/tex]
This is a quadratic equation since it is in the form [tex]\(ax^2 + bx + c = 0\)[/tex] with [tex]\(a = 5\)[/tex], [tex]\(b = 0\)[/tex], and [tex]\(c = -90\)[/tex].
### Option d) [tex]\( 2x - x^2 = x^2 + 5 \)[/tex]
1. Bring all terms to one side to set the equation to zero:
[tex]\[ 2x - x^2 - x^2 = 5 \][/tex]
[tex]\[ -2x^2 + 2x = 5 \][/tex]
2. Rearrange to standard quadratic form:
[tex]\[ -2x^2 + 2x - 5 = 0 \][/tex]
This is a quadratic equation since it is in the form [tex]\(ax^2 + bx + c = 0\)[/tex] with [tex]\(a = -2\)[/tex], [tex]\(b = 2\)[/tex], and [tex]\(c = -5\)[/tex].
### Conclusion:
Out of the given options, only option [tex]\( a) \)[/tex] does not form a quadratic equation after simplification. Thus, the correct answer is:
a) [tex]\( x^2 + 4x = 11 + x^2 \)[/tex]
