Given x^2 - 4x + 9 = (x+A)^2 + B.
This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
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Step 1: Expand the right side of the equation.
Given $x^2 - 4x + 9 = (x+A)^2 + B$.
Expand $(x+A)^2$:
$$(x+A)^2 = x^2 + 2Ax + A^2$$
So, the right side becomes:
$$x^2 + 2Ax + A^2 + B$$
Step 2: Compare coefficients.
Equate the coefficients of the expanded right side with the left side $x^2 - 4x + 9$:
For the $x^2$ term: $1 = 1$ (matches)
For the $x$ term: $2A = -4$
For the constant term: $A^2 + B = 9$
Step 3: Solve for A and B.
From $2A = -4$:
$$A = \frac{-4}{2} = -2$$
Substitute $A=-2$ into $A^2 + B = 9$:
$$(-2)^2 + B = 9$$
$$4 + B = 9$$
$$B = 9 - 4$$
$$B = 5$$
Thus, $A=-2$ and $B=5$.
The correct option is D.
The final answer is $\boxed{\text{D}}$.
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