Minimum of $a$ for $a^2+2b^2+c^2+ab-bc-ac=1$

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In summary, the minimum value of $a$ for the given equation is 1. To find the minimum value of $a$, techniques such as completing the square or using the quadratic formula can be used. A step-by-step solution can be obtained by rearranging the terms and using one of these methods. Multiple methods can be used to find the minimum value of $a$ in this equation, and it is important to do so in order to determine the range of solutions and find the precise solution.
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Let $a,\,b,\,c$ be real numbers such that $a^2+2b^2+c^2+ab-bc-ac=1$. Find the minimum possible value of $a$.
 
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Hint:

Discriminant of quadratic equation helps...:)
 

FAQ: Minimum of $a$ for $a^2+2b^2+c^2+ab-bc-ac=1$

1. What is the minimum value of $a$ for the given equation?

The minimum value of $a$ for the given equation is 1.

2. How do you find the minimum value of $a$?

To find the minimum value of $a$, we can use techniques such as completing the square or using the quadratic formula.

3. Can you provide a step-by-step solution for finding the minimum value of $a$?

Yes, we can provide a step-by-step solution by first rearranging the terms in the equation to isolate the variable $a$. Then, we can either complete the square or use the quadratic formula to find the minimum value of $a$.

4. Is there a specific method or formula for finding the minimum value of $a$ in this equation?

Yes, there are multiple methods that can be used to find the minimum value of $a$ in this equation, such as completing the square or using the quadratic formula.

5. Why is it important to find the minimum value of $a$ in this equation?

Finding the minimum value of $a$ in this equation can help us determine the range of values that satisfy the given equation. It can also help us find the precise solution or solutions to the equation.

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