Write x² + 6x + 11 in the form (x + a)² + b where a and b are integers.
Understand the Problem
The question is asking to rewrite the expression x² + 6x + 11 in the form (x + a)² + b, where a and b are integers. This requires completing the square for the quadratic expression.
Answer
The expression is $(x + 3)^2 + 2$.
Answer for screen readers
The expression $x^2 + 6x + 11$ can be rewritten as $(x + 3)^2 + 2$.
Steps to Solve
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Identify the quadratic expression We have the expression $x^2 + 6x + 11$. To rewrite it in the form $(x + a)^2 + b$, we need to focus on completing the square.
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Complete the square on the first two terms Take the coefficient of $x$, which is 6, divide it by 2, and square it: $$ \left(\frac{6}{2}\right)^2 = 3^2 = 9 $$
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Rewrite the expression using the completed square Now we can use this value to express the quadratic: $$ x^2 + 6x = (x + 3)^2 - 9 $$ So the entire expression becomes: $$ (x + 3)^2 - 9 + 11 $$
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Simplify the expression Now combine the constants: $$ -9 + 11 = 2 $$ Thus, the expression simplifies to: $$ (x + 3)^2 + 2 $$
The expression $x^2 + 6x + 11$ can be rewritten as $(x + 3)^2 + 2$.
More Information
Completing the square is a useful technique in algebra, especially for graphing quadratic functions and solving quadratic equations. The vertex form of a quadratic function allows us to easily determine its vertex and direction.
Tips
- Failing to properly complete the square by forgetting to adjust the constant term.
- Miscalculating the square of half the coefficient of $x$.
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