Write x² + 6x + 13 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 + 6x + 13 in the form of (x + a)² + b, where a and b are integers. This involves completing the square.
Answer
$(x + 3)^2 + 4$
Answer for screen readers
The expression $x^2 + 6x + 13$ in the form $(x + a)^2 + b$ is $(x + 3)^2 + 4$ where $a = 3$ and $b = 4$.
Steps to Solve
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Identify the quadratic expression The quadratic expression given is $x^2 + 6x + 13$.
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Complete the square To complete the square, take half of the coefficient of $x$, which is $6$. Half of $6$ is $3$, and squaring it gives $3^2 = 9$.
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Rewrite the expression Add and subtract $9$ inside the expression: $$x^2 + 6x + 9 - 9 + 13$$
This simplifies to: $$ (x^2 + 6x + 9) + 4 $$
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Factor the perfect square Now, recognize that $x^2 + 6x + 9$ is a perfect square: $$ (x + 3)^2 + 4 $$
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Identify integers a and b In the form $(x + a)^2 + b$, we have $a = 3$ and $b = 4$.
The expression $x^2 + 6x + 13$ in the form $(x + a)^2 + b$ is $(x + 3)^2 + 4$ where $a = 3$ and $b = 4$.
More Information
Completing the square is a useful technique not only for simplifying quadratic expressions but also for solving quadratic equations and understanding the properties of parabolas.
Tips
- Forgetting to balance the equation when adding and subtracting the square term.
- Not recognizing the perfect square trinomial when rewriting it.
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