The Roots Of The Equation $2x^2 + (a+1)x + B = 0$ Are 1 And 3, Where $a$ And $ B B B [/tex] Are Constants. Find The Values Of $a$ And $b$.7. The Equation $ax^2 + Bx + C = 0$ Has Roots

Mar 8, 2025 by ADMIN 194 views

6. The Roots of a Quadratic Equation: Finding the Values of Constants

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Introduction

In this article, we will explore the concept of finding the values of constants in a quadratic equation given its roots. 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.

The Given Equation

The given equation is $2x^2 + (a+1)x + b = 0$, and its roots are 1 and 3. This means that when we substitute $x = 1$ and $x = 3$ into the equation, it should satisfy the equation.

Substituting the Roots

Let's substitute $x = 1$ into the equation:

2(1)^2 + (a+1)(1) + b = 0

Simplifying the equation, we get:

2 + a + 1 + b = 0

Combine like terms:

a + b + 3 = 0

Now, let's substitute $x = 3$ into the equation:

2(3)^2 + (a+1)(3) + b = 0

Simplifying the equation, we get:

18 + 3a + 3 + b = 0

Combine like terms:

3a + b + 21 = 0

Solving the System of Equations

We now have a system of two equations with two variables:

a + b + 3 = 0

3a + b + 21 = 0

We can solve this system of equations using substitution or elimination. Let's use the elimination method.

Elimination Method

Subtract the first equation from the second equation:

(3a + b + 21) - (a + b + 3) = 0 - 0

Simplifying the equation, we get:

2a + 18 = 0

Subtract 18 from both sides:

2a = -18

Divide both sides by 2:

a = -9

Finding the Value of b

Now that we have the value of $a$, we can substitute it into one of the original equations to find the value of $b$. Let's use the first equation:

a + b + 3 = 0

Substitute $a = -9$:

-9 + b + 3 = 0

Simplify the equation:

b - 6 = 0

Add 6 to both sides:

b = 6

Conclusion

In this article, we found the values of $a$ and $b$ in the quadratic equation $2x^2 + (a+1)x + b = 0$ given its roots 1 and 3. We used the substitution method to find the values of $a$ and $b$.

The Final Answer

The final answer is:

a = -9

b = 6

Discussion

The equation $ax^2 + bx + c = 0$ has roots 1 and 3. We can use the same method to find the values of $a$, $b$, and $c$ in this equation.

Example

Let's consider the equation $x^2 + (a+1)x + b = 0$ with roots 1 and 3. We can follow the same steps as before to find the values of $a$ and $b$.

Solution

Substitute $x = 1$ into the equation:

1^2 + (a+1)(1) + b = 0

Simplify the equation:

1 + a + 1 + b = 0

Combine like terms:

a + b + 2 = 0

Now, let's substitute $x = 3$ into the equation:

3^2 + (a+1)(3) + b = 0

Simplify the equation:

9 + 3a + 3 + b = 0

Combine like terms:

3a + b + 12 = 0

Solving the System of Equations

We now have a system of two equations with two variables:

a + b + 2 = 0

3a + b + 12 = 0

We can solve this system of equations using substitution or elimination. Let's use the elimination method.

Elimination Method

Subtract the first equation from the second equation:

(3a + b + 12) - (a + b + 2) = 0 - 0

Simplify the equation:

2a + 10 = 0

Subtract 10 from both sides:

2a = -10

Divide both sides by 2:

a = -5

Finding the Value of b

Now that we have the value of $a$, we can substitute it into one of the original equations to find the value of $b$. Let's use the first equation:

a + b + 2 = 0

Substitute $a = -5$:

-5 + b + 2 = 0

Simplify the equation:

b - 3 = 0

Add 3 to both sides:

b = 3

Conclusion

In this article, we found the values of $a$ and $b$ in the quadratic equation $x^2 + (a+1)x + b = 0$ given its roots 1 and 3. We used the substitution method to find the values of $a$ and $b$.

The Final Answer

The final answer is:

a = -5

b = 3$<br/> # 7. Frequently Asked Questions (FAQs) About Finding the Values of Constants in a Quadratic Equation

Introduction

In the previous article, we explored the concept of finding the values of constants in a quadratic equation given its roots. In this article, we will answer some frequently asked questions (FAQs) about this topic.

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 $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $x$ is the variable.

Q: How do I find the values of constants in a quadratic equation?

A: To find the values of constants in a quadratic equation, you need to know the roots of the equation. You can use the substitution method or the elimination method to find the values of constants.

Q: What is the substitution method?

A: The substitution method is a method of solving a system of equations by substituting one equation into another equation. In the context of finding the values of constants in a quadratic equation, you substitute the roots of the equation into the equation to find the values of constants.

Q: What is the elimination method?

A: The elimination method is a method of solving a system of equations by eliminating one variable. In the context of finding the values of constants in a quadratic equation, you eliminate one variable by subtracting one equation from another equation.

Q: How do I know which method to use?

A: You can use either the substitution method or the elimination method to find the values of constants in a quadratic equation. The choice of method depends on the specific equation and the roots of the equation.

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

A: If you have a quadratic equation with complex roots, you can use the same methods to find the values of constants. However, you need to be careful when working with complex numbers.

Q: Can I use a calculator to find the values of constants in a quadratic equation?

A: Yes, you can use a calculator to find the values of constants in a quadratic equation. However, it's always a good idea to check your work by hand to make sure you get the correct answer.

Q: What if I have a quadratic equation with no real roots?

A: If you have a quadratic equation with no real roots, it means that the equation has complex roots. You can use the same methods to find the values of constants, but you need to be careful when working with complex numbers.

Conclusion

In this article, we answered some frequently asked questions (FAQs) about finding the values of constants in a quadratic equation. We hope this article has been helpful in clarifying any doubts you may have had about this topic.

The Final Answer

The final answer is:

The substitution method and the elimination method are two methods of finding the values of constants in a quadratic equation.
The choice of method depends on the specific equation and the roots of the equation.
You can use a calculator to find the values of constants in a quadratic equation, but it's always a good idea to check your work by hand to make sure you get the correct answer.

Discussion

The equation $ax^2 + bx + c = 0$ has roots 1 and 3. We can use the same methods to find the values of $a$, $b$, and $c$ in this equation.

Example

Let's consider the equation $x^2 + (a+1)x + b = 0$ with roots 1 and 3. We can follow the same steps as before to find the values of $a$ and $b$.

Solution

Substitute $x = 1$ into the equation: