Simplify the expression $-3(x+3)^2-3+3x$. What is the simplified expression in standard form?
The problem asks to simplify the quadratic expression \(-3(x+3)^2 - 3 + 3x\) and write it in the standard form \(Ax^2 + Bx + C\). To do this, we follow the order of operations, starting with the exponentiation (expanding the squared binomial). Then we apply the distributive property to multiply the expanded expression by the coefficient \(-3\), and finally, we combine like terms.
First, expand the squared term \((x+3)^2\) using the binomial square formula \((a+b)^2 = a^2 + 2ab + b^2\). Here, \(a = x\) and \(b = 3\):
\[(x+3)^2 = x^2 + 2(x)(3) + 3^2 = x^2 + 6x + 9.\]
Now, substitute this back into the original expression:
\[-3(x^2 + 6x + 9) - 3 + 3x.\]
Apply the distributive property by multiplying \(-3\) by each term inside the parentheses:
\[-3(x^2 + 6x + 9) = -3x^2 - 18x - 27.\]
The full expression now becomes:
\[-3x^2 - 18x - 27 - 3 + 3x.\]
Group the like terms together to simplify the expression further:
\[-3x^2 + (-18x + 3x) + (-27 - 3).\]
Now, perform the addition and subtraction for the coefficients:
\[-18x + 3x = -15x\]
\[-27 - 3 = -30\]
So, the final standard form of the expression is \(-3x^2 - 15x - 30\). Comparing this result with the given options, we identify the correct choice. The simplified expression is \(-3x^2 - 15x - 30\).
Understanding the expansion and simplification of quadratic expressions is essential in fields like physics and engineering. For example, if the trajectory of a projectile is modeled by an equation like \(-3(t+3)^2 - 3 + 3t\), simplifying this allows engineers to easily find the time \(t\) when the projectile hits the ground (by setting the quadratic to zero) or determine the vertex of its path. Proficiency with these algebraic manipulations is a building block for more complex modeling.