Common roots and finding real values

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In summary, common roots are values that satisfy a polynomial equation and can be found using methods such as factoring, the quadratic formula, or synthetic division. It is important to find common roots because it allows us to solve equations and understand the behavior of a polynomial function. Real values are solutions that can be graphed on a number line, while imaginary values are represented by the square root of a negative number and cannot be graphed.
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Sudharaka
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srirahulan's question titled "Algeb" from Math Help Forum,

If \(ax^2+2x+1=0\mbox{ and }x^2+2x+a=0\) have the common roots, find the real value of a.

Hi srirahulan,

Let \(\alpha\mbox{ and }\beta\) be the two roots of these quadratic equations. Then, according to the first equation,

\[\alpha+\beta=-\frac{2}{a}~~~~~~(1)\]

Considering the second equation,

\[\alpha+\beta=-2~~~~~~~(2)\]

By (1) and (2);

\[-\frac{2}{a}=-2\]

\[\therefore a=1\]
 
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What is the definition of common roots?

Common roots refer to the values that satisfy a given polynomial equation, meaning they are the values that make the equation true when plugged in. In other words, they are the solutions to the equation.

How do you find common roots?

To find common roots, you can use methods such as factoring, the quadratic formula, or synthetic division. These methods help simplify the polynomial equation and make it easier to determine the values that make it true.

Why is it important to find common roots?

Finding common roots is important because it allows us to solve equations and understand the behavior of a polynomial function. It also helps us identify the x-intercepts, or the points where the graph of the polynomial crosses the x-axis.

What are real values?

Real values refer to the solutions of an equation that are part of the set of real numbers. They are the numbers that can be graphed on a number line and have a clear position in relation to other numbers.

How do you determine if a value is real or imaginary?

If a value satisfies a polynomial equation and can be represented on the number line, it is a real value. If a value does not satisfy the equation and is represented by the square root of a negative number, it is an imaginary value. Imaginary values cannot be graphed on a number line.

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