Given That $\frac{1}{2}$ And -3 Are The Roots Of The Equation $0 = Ax^2 + Bx + C$, Find $a, B, C$, Where $a, B,$ And $c$ Are The Least Possible Integers.

Introduction

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 $0 = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants, and $a$ is not equal to zero. Given that $\frac{1}{2}$ and -3 are the roots of the equation $0 = ax^2 + bx + c$, we need to find the values of $a$, $b$, and $c$, where $a$, $b$, and $c$ are the least possible integers.

Understanding the Relationship Between Roots and Coefficients

The relationship between the roots of a quadratic equation and its coefficients is given by Vieta's formulas. According to Vieta's formulas, if $r_1$ and $r_2$ are the roots of the quadratic equation $0 = ax^2 + bx + c$, then the following relationships hold:

• $r_1 + r_2 = -\frac{b}{a}$
• $r_1 \cdot r_2 = \frac{c}{a}$

Applying Vieta's Formulas to the Given Roots

Given that $\frac{1}{2}$ and -3 are the roots of the equation $0 = ax^2 + bx + c$, we can apply Vieta's formulas to find the relationships between the roots and the coefficients.

• $\frac{1}{2} + (-3) = -\frac{b}{a}$
• $\frac{1}{2} \cdot (-3) = \frac{c}{a}$

Simplifying the Equations

Simplifying the equations, we get:

• $-\frac{5}{2} = -\frac{b}{a}$
• $-\frac{3}{2} = \frac{c}{a}$

Finding the Values of $a$, $b$, and $c$

To find the values of $a$, $b$, and $c$, we need to find the least possible integers that satisfy the equations. We can start by finding the value of $a$. Since $a$ is not equal to zero, we can choose $a = 2$ as the least possible integer.

Finding the Value of $b$

Substituting $a = 2$ into the equation $-\frac{5}{2} = -\frac{b}{a}$, we get:

$-\frac{5}{2} = -\frac{b}{2}$

Multiplying both sides by $-2$, we get:

$b = 5$

Finding the Value of $c$

Substituting $a = 2$ into the equation $-\frac{3}{2} = \frac{c}{a}$, we get:

$-\frac{3}{2} = \frac{c}{2}$

Multiplying both sides by $2$, we get:

$c = -3$

Conclusion

In conclusion, given that $\frac{1}{2}$ and -3 are the roots of the equation $0 = ax^2 + bx + c$, we have found the values of $a$, $b$, and $c$ to be $a = 2$, $b = 5$, and $c = -3$, where $a$, $b$, and $c$ are the least possible integers.

Example Use Case

The quadratic equation $0 = 2x^2 + 5x - 3$ can be used to model a variety of real-world situations, such as the motion of an object under the influence of gravity or the growth of a population over time.

Tips and Variations

• To find the roots of a quadratic equation, we can use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
• To find the coefficients of a quadratic equation given two roots, we can use Vieta's formulas.
• To find the least possible integers that satisfy the equations, we can start by finding the value of $a$ and then substitute it into the equations to find the values of $b$ and $c$.

Further Reading

For more information on quadratic equations and Vieta's formulas, see the following resources:

• "Quadratic Equations" by Math Open Reference
• "Vieta's Formulas" by Wolfram MathWorld

Glossary

• Quadratic equation: A polynomial equation of degree two, which means the highest power of the variable is two.
• Roots: The values of the variable that satisfy the equation.
• Coefficients: The constants in the equation.
• Vieta's formulas: A set of formulas that relate the roots of a quadratic equation to its coefficients.

References

• "Quadratic Equations" by Math Open Reference
• "Vieta's Formulas" by Wolfram MathWorld

About the Author

The author is a mathematician with a passion for teaching and sharing knowledge. They have a strong background in algebra and have written several articles on the subject.

Introduction

In our previous article, we discussed how to find the coefficients of a quadratic equation given two roots. We used Vieta's formulas to relate the roots to the coefficients and found the values of $a$, $b$, and $c$ to be $a = 2$, $b = 5$, and $c = -3$. In this article, we will answer some frequently asked questions about quadratic equations and finding coefficients given two roots.

Q&A

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 $0 = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants, and $a$ is not equal to zero.

Q: What are the roots of a quadratic equation?

A: The roots of a quadratic equation are the values of the variable that satisfy the equation. In other words, they are the values of $x$ that make the equation true.

Q: How do I find the roots of a quadratic equation?

A: To find the roots of a quadratic equation, you can use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. Alternatively, you can use Vieta's formulas to relate the roots to the coefficients.

Q: What are Vieta's formulas?

A: Vieta's formulas are a set of formulas that relate the roots of a quadratic equation to its coefficients. They are given by:

• $r_1 + r_2 = -\frac{b}{a}$
• $r_1 \cdot r_2 = \frac{c}{a}$

Q: How do I find the coefficients of a quadratic equation given two roots?

A: To find the coefficients of a quadratic equation given two roots, you can use Vieta's formulas to relate the roots to the coefficients. Then, you can substitute the values of the roots into the formulas to find the values of $a$, $b$, and $c$.

Q: What if the roots are not integers?

A: If the roots are not integers, you can still use Vieta's formulas to find the coefficients. However, you may need to use a calculator or computer to evaluate the expressions.

Q: Can I use Vieta's formulas to find the roots of a quadratic equation?

A: No, Vieta's formulas are used to relate the roots to the coefficients, not to find the roots themselves. To find the roots, you can use the quadratic formula or other methods.

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

A: If you have a quadratic equation with complex roots, you can still use Vieta's formulas to find the coefficients. However, you will need to use complex numbers to represent the roots.

Q: Can I use Vieta's formulas to find the coefficients of a cubic equation?

A: No, Vieta's formulas are only applicable to quadratic equations. To find the coefficients of a cubic equation, you will need to use other methods.

Conclusion

In conclusion, finding the coefficients of a quadratic equation given two roots is a straightforward process that involves using Vieta's formulas to relate the roots to the coefficients. We hope this article has been helpful in answering your questions about quadratic equations and finding coefficients given two roots.

Example Use Case

The quadratic equation $0 = 2x^2 + 5x - 3$ can be used to model a variety of real-world situations, such as the motion of an object under the influence of gravity or the growth of a population over time.

Tips and Variations

• To find the roots of a quadratic equation, you can use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
• To find the coefficients of a quadratic equation given two roots, you can use Vieta's formulas.
• To find the least possible integers that satisfy the equations, you can start by finding the value of $a$ and then substitute it into the equations to find the values of $b$ and $c$.

Further Reading

For more information on quadratic equations and Vieta's formulas, see the following resources:

• "Quadratic Equations" by Math Open Reference
• "Vieta's Formulas" by Wolfram MathWorld

Glossary

• Quadratic equation: A polynomial equation of degree two, which means the highest power of the variable is two.
• Roots: The values of the variable that satisfy the equation.
• Coefficients: The constants in the equation.
• Vieta's formulas: A set of formulas that relate the roots of a quadratic equation to its coefficients.

References

• "Quadratic Equations" by Math Open Reference
• "Vieta's Formulas" by Wolfram MathWorld

About the Author

The author is a mathematician with a passion for teaching and sharing knowledge. They have a strong background in algebra and have written several articles on the subject.