Write x² - 6x + 16 in the form (x + a)² + b where a and b are integers.
Understand the Problem
The question asks to reformat the quadratic equation x² - 6x + 16 into the form (x + a)² + b, where a and b are integers. This involves completing the square for the given quadratic expression.
Answer
The equation is \( (x - 3)^2 + 7 \).
Answer for screen readers
The expression in the desired form is ( (x - 3)^2 + 7 ).
Steps to Solve
- Identify the quadratic expression
We have the quadratic expression ( x^2 - 6x + 16 ).
- Complete the square for the (x) terms
To complete the square, take the coefficient of (x) (which is (-6)), divide it by (2) to get (-3), and square it to obtain (9).
- Rewrite the expression
Rewriting the quadratic expression, we can express it as:
$$ x^2 - 6x + 9 + 16 - 9 $$
This simplifies to:
$$ (x - 3)^2 + 7 $$
- Final expression
Thus, we can rewrite the original quadratic expression as:
$$ (x - 3)^2 + 7 $$
Therefore, (a = -3) and (b = 7).
The expression in the desired form is ( (x - 3)^2 + 7 ).
More Information
This method of completing the square helps in rewriting quadratic equations, making them easier to analyze, especially for finding vertex form or solving for roots.
Tips
- Forgetting to adjust the constant: When completing the square, remember to add and subtract the squared term to keep the equation balanced.
- Incorrect sign for (a): Ensure the transformed term matches the original quadratic coefficient.
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