To determine which of the given equations is quadratic in form, we need to identify the equation that can be written in the standard quadratic form [tex]\( ax^2 + bx + c = 0 \)[/tex]. Let's look at each equation step by step.
1. Equation: [tex]\( 4(x-2)^2 + 3x - 2 + 1 = 0 \)[/tex]
First, we expand [tex]\( 4(x-2)^2 \)[/tex]:
[tex]\[
4(x-2)^2 = 4(x^2 - 4x + 4) = 4x^2 - 16x + 16
\][/tex]
Next, we combine the expanded equation with the remaining terms:
[tex]\[
4x^2 - 16x + 16 + 3x - 2 + 1 = 0
\][/tex]
Simplify the equation:
[tex]\[
4x^2 - 16x + 3x + 16 - 2 + 1 = 0
\][/tex]
[tex]\[
4x^2 - 13x + 15 = 0
\][/tex]
This equation is of the form [tex]\( ax^2 + bx + c = 0 \)[/tex], which makes it a quadratic equation.
2. Equation: [tex]\( 8x^5 + 4x^3 + 1 = 0 \)[/tex]
This equation involves [tex]\( x^5 \)[/tex] and [tex]\( x^3 \)[/tex] terms. Since a quadratic equation specifically requires an [tex]\( x^2 \)[/tex] term as the highest degree term, this equation is not quadratic.
3. Equation: [tex]\( 10x^8 + 7x^4 + 1 = 0 \)[/tex]
This equation involves [tex]\( x^8 \)[/tex] and [tex]\( x^4 \)[/tex] terms. With the highest degree term being [tex]\( x^8 \)[/tex], this equation is not quadratic.
4. Equation: [tex]\( 9x^{16} + 6x^4 + 1 = 0 \)[/tex]
This equation involves [tex]\( x^{16} \)[/tex] and [tex]\( x^4 \)[/tex] terms. With the highest degree term being [tex]\( x^{16} \)[/tex], this equation is not quadratic.
From the analysis, we can see that the first equation, [tex]\( 4(x-2)^2 + 3x - 2 + 1 = 0 \)[/tex], simplifies to [tex]\( 4x^2 - 13x + 15 = 0 \)[/tex], which is a quadratic equation. Therefore, the equation that is quadratic in form is:
[tex]\[ \boxed{4(x-2)^2 + 3x - 2 + 1 = 0} \][/tex]
or
[tex]\[ \boxed{1}. \][/tex]
