Given The Roots { -1$}$ And { \frac{3}{2}$}$ For The Equation 2 X 2 + B X + C = 0 2x^2 + Bx + C = 0 2 X 2 + B X + C = 0 , Find { B$}$ And { C$}$.

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Introduction

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In algebra, a quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $x$ is the variable. Given the roots of a quadratic equation, we can find the values of $b$ and $c$ using the relationships between the roots and the coefficients of the equation.

The Relationship Between Roots and Coefficients

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The roots of a quadratic equation $ax^2 + bx + c = 0$ are the values of $x$ that satisfy the equation. If the roots are denoted by $r_1$ and $r_2$, then the equation can be factored as $a(x - r_1)(x - r_2) = 0$. Expanding this expression, we get $a(x^2 - (r_1 + r_2)x + r_1r_2) = 0$. Comparing this with the original equation, we see that $b = -a(r_1 + r_2)$ and $c = ar_1r_2$.

Finding the Coefficients Given the Roots

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Given the roots $r_1 = -1$ and $r_2 = \frac{3}{2}$, we can find the values of $b$ and $c$ using the relationships derived above. First, we need to find the value of $a$. Since the equation is $2x^2 + bx + c = 0$, we can see that $a = 2$.

Calculating the Value of b

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Using the formula $b = -a(r_1 + r_2)$, we can substitute the values of $a$, $r_1$, and $r_2$ to find the value of $b$. We have $b = -2(-1 + \frac{3}{2}) = -2(-\frac{1}{2}) = 1$.

Calculating the Value of c

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Using the formula $c = ar_1r_2$, we can substitute the values of $a$, $r_1$, and $r_2$ to find the value of $c$. We have $c = 2(-1)(\frac{3}{2}) = -3$.

Conclusion

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In this article, we have shown how to find the coefficients $b$ and $c$ of a quadratic equation given its roots. We have used the relationships between the roots and the coefficients to derive formulas for $b$ and $c$. By substituting the values of the roots into these formulas, we have found the values of $b$ and $c$ for the given equation.

Example Use Case

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Suppose we are given the roots $r_1 = 2$ and $r_2 = -3$ for the equation $x^2 + bx + c = 0$. We can use the formulas derived above to find the values of $b$ and $c$. First, we need to find the value of $a$. Since the equation is $x^2 + bx + c = 0$, we can see that $a = 1$. Then, we can use the formula $b = -a(r_1 + r_2)$ to find the value of $b$. We have $b = -1(2 + (-3)) = -1(-1) = 1$. Finally, we can use the formula $c = ar_1r_2$ to find the value of $c$. We have $c = 1(2)(-3) = -6$.

Final Answer

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To summarize, given the roots $r_1 = -1$ and $r_2 = \frac{3}{2}$ for the equation $2x^2 + bx + c = 0$, we have found the values of $b$ and $c$ to be $b = 1$ and $c = -3$.

References

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• [1] "Quadratic Equation" by Math Open Reference. Retrieved from https://www.mathopenref.com/quadratic.html
• [2] "Roots of a Quadratic Equation" by Purplemath. Retrieved from https://www.purplemath.com/modules/quadroots.htm

Future Work

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In future work, we can explore other methods for finding the coefficients of a quadratic equation given its roots. For example, we can use the quadratic formula to find the roots of the equation, and then use the relationships between the roots and the coefficients to find the values of $b$ and $c$. We can also investigate the relationship between the coefficients and the roots of a quadratic equation in more detail, and explore the implications of this relationship for solving quadratic equations.

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Introduction

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In our previous article, we discussed how to find the coefficients $b$ and $c$ of a quadratic equation given its roots. In this article, we will provide a Q&A guide to help you understand the concepts and formulas involved in finding the coefficients of a quadratic equation.

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $x$ is the variable.

Q: What are the roots of a quadratic equation?

A: The roots of a quadratic equation are the values of $x$ that satisfy the equation. If the roots are denoted by $r_1$ and $r_2$, then the equation can be factored as $a(x - r_1)(x - r_2) = 0$. Expanding this expression, we get $a(x^2 - (r_1 + r_2)x + r_1r_2) = 0$.

Q: How do I find the coefficients $b$ and $c$ given the roots?

A: To find the coefficients $b$ and $c$ given the roots, you can use the formulas $b = -a(r_1 + r_2)$ and $c = ar_1r_2$. First, you need to find the value of $a$ by looking at the equation. Then, you can substitute the values of $a$, $r_1$, and $r_2$ into the formulas to find the values of $b$ and $c$.

Q: What if I have a quadratic equation in the form $x^2 + bx + c = 0$?

A: If you have a quadratic equation in the form $x^2 + bx + c = 0$, then you can use the formulas $b = -(r_1 + r_2)$ and $c = r_1r_2$ to find the values of $b$ and $c$ given the roots. First, you need to find the value of $a$ by looking at the equation. Then, you can substitute the values of $a$, $r_1$, and $r_2$ into the formulas to find the values of $b$ and $c$.

Q: Can I use the quadratic formula to find the roots of the equation?

A: Yes, you can use the quadratic formula to find the roots of the equation. The quadratic formula is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. However, if you are given the roots, it is often easier to use the formulas $b = -a(r_1 + r_2)$ and $c = ar_1r_2$ to find the values of $b$ and $c$.

Q: What if I have a quadratic equation with complex roots?

A: If you have a quadratic equation with complex roots, then you can use the formulas $b = -a(r_1 + r_2)$ and $c = ar_1r_2$ to find the values of $b$ and $c$ given the roots. However, you will need to use complex numbers to represent the roots.

Q: Can I use the quadratic formula to find the coefficients $b$ and $c$?

A: No, you cannot use the quadratic formula to find the coefficients $b$ and $c$. The quadratic formula is used to find the roots of the equation, not the coefficients.

Q: What if I have a quadratic equation with a coefficient of $a = 0$?

A: If you have a quadratic equation with a coefficient of $a = 0$, then the equation is not quadratic. In this case, you can use the formulas $b = -(r_1 + r_2)$ and $c = r_1r_2$ to find the values of $b$ and $c$ given the roots.

Q: Can I use the formulas to find the coefficients $b$ and $c$ for any quadratic equation?

A: Yes, you can use the formulas to find the coefficients $b$ and $c$ for any quadratic equation. However, you will need to make sure that the equation is in the form $ax^2 + bx + c = 0$.

Q: What if I have a quadratic equation with a coefficient of $b = 0$?

A: If you have a quadratic equation with a coefficient of $b = 0$, then the equation is of the form $ax^2 + c = 0$. In this case, you can use the formula $c = ar_1r_2$ to find the value of $c$ given the roots.

Q: Can I use the formulas to find the coefficients $b$ and $c$ for a quadratic equation with a coefficient of $c = 0$?

A: No, you cannot use the formulas to find the coefficients $b$ and $c$ for a quadratic equation with a coefficient of $c = 0$. In this case, the equation is of the form $ax^2 + bx = 0$, and you will need to use a different method to find the values of $b$ and $c$.

Conclusion

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In this Q&A guide, we have provided answers to common questions about finding the coefficients $b$ and $c$ of a quadratic equation given its roots. We have also discussed the formulas and methods involved in finding the coefficients, and provided examples to illustrate the concepts. We hope that this guide has been helpful in understanding the concepts and formulas involved in finding the coefficients of a quadratic equation.