Write x² - 2x + 3 in the form (x + a)² + b where a and b are integers.
Understand the Problem
The question is asking to rewrite the quadratic expression x² - 2x + 3 in the format of (x + a)² + b, where a and b are integers. This involves completing the square.
Answer
The expression is \((x - 1)^2 + 2\) with \(a = -1\) and \(b = 2\).
Answer for screen readers
The expression in the desired form is ((x - 1)^2 + 2) where (a = -1) and (b = 2).
Steps to Solve
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Identify the quadratic expression We need to rewrite the quadratic expression (x^2 - 2x + 3) in the form ((x + a)^2 + b).
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Complete the square To complete the square, take the coefficient of (x) (which is -2), divide it by 2, and square the result. $$ \left(\frac{-2}{2}\right)^2 = 1 $$
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Rewrite the expression Now, rewrite the expression by adding and subtracting this square inside the expression: $$ x^2 - 2x + 1 - 1 + 3 $$ This simplifies to: $$ (x - 1)^2 + 2 $$
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Identify values of (a) and (b) From the rewritten expression ((x - 1)^2 + 2), we can identify (a) and (b):
- (a = -1)
- (b = 2)
The expression in the desired form is ((x - 1)^2 + 2) where (a = -1) and (b = 2).
More Information
Completing the square is a powerful technique to rewrite quadratic expressions, which can help in graphing and solving quadratic equations. The vertex form of a quadratic can be particularly useful for understanding the properties of the function, such as its maximum or minimum value.
Tips
- Forgetting to keep the balance when adding and subtracting the square: Always add and subtract the same value.
- Miscomputing the value obtained from dividing the coefficient of (x) by 2 and squaring it: Ensure correct arithmetic is done.
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