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Given x² − 4x + 9 = (x+A)² + B.

April 14, 2026Mathematics

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Answer

3 steps

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:
$x^2 + 2Ax + A^2 + B$
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: $A^2 + B = 9$
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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