To transform the quadratic equation [tex]\(y = (2x + 3)^2\)[/tex] into its standard form, we need to expand the expression [tex]\((2x + 3)^2\)[/tex].
Here are the steps to do this:
1. Start with the given equation:
[tex]\[ y = (2x + 3)^2 \][/tex]
2. To expand [tex]\((2x + 3)^2\)[/tex], use the binomial theorem [tex]\((a + b)^2 = a^2 + 2ab + b^2\)[/tex], where [tex]\(a = 2x\)[/tex] and [tex]\(b = 3\)[/tex].
3. Apply the binomial theorem:
[tex]\[ (2x + 3)^2 = (2x)^2 + 2 \cdot (2x) \cdot 3 + 3^2 \][/tex]
4. Calculate each term:
[tex]\[ (2x)^2 = 4x^2 \][/tex]
[tex]\[ 2 \cdot (2x) \cdot 3 = 12x \][/tex]
[tex]\[ 3^2 = 9 \][/tex]
5. Combine all the terms:
[tex]\[ (2x + 3)^2 = 4x^2 + 12x + 9 \][/tex]
6. Set [tex]\(y = (2x + 3)^2\)[/tex] to [tex]\(0\)[/tex] to find the standard form of the quadratic equation:
[tex]\[ y = 4x^2 + 12x + 9 \][/tex]
Therefore, the standard form of the given quadratic equation is:
[tex]\[ 0 = 4x^2 + 12x + 9 \][/tex]
So, the correct choice is:
[tex]\[ 0 = 4 x^2 + 12 x + 9 \][/tex]
