Find The Values Of $a$ And $b$ Such That ${x^2 + 2x - 7 = (x + A)^2 + B}$

Introduction

In this article, we will delve into the world of quadratic equations and explore a method to find the values of $a$ and $b$ in the given equation. The equation in question is $x^2 + 2x - 7 = (x + a)^2 + b$. Our goal is to determine the values of $a$ and $b$ that satisfy this equation.

Understanding the Equation

The given equation is a quadratic equation in the form of $x^2 + 2x - 7 = (x + a)^2 + b$. To begin solving this equation, we need to expand the right-hand side of the equation. The expanded form of $(x + a)^2$ is $x^2 + 2ax + a^2$. Substituting this into the original equation, we get:

$x^2 + 2x - 7 = x^2 + 2ax + a^2 + b$

Simplifying the Equation

Now that we have expanded the right-hand side of the equation, we can simplify it by combining like terms. The $x^2$ terms on both sides of the equation cancel each other out, leaving us with:

$2x - 7 = 2ax + a^2 + b$

Isolating the Terms

Our next step is to isolate the terms containing $a$ and $b$ on one side of the equation. To do this, we can subtract $2ax$ from both sides of the equation and then subtract $a^2$ from both sides. This gives us:

$2x - 7 - 2ax = a^2 + b$

Rearranging the Terms

We can further simplify the equation by rearranging the terms. Let's move the $a^2$ term to the left-hand side of the equation and the $b$ term to the right-hand side. This gives us:

$2x - 7 - 2ax - a^2 = b$

Simplifying the Left-Hand Side

Now that we have rearranged the terms, we can simplify the left-hand side of the equation. We can combine the $2x$ and $-2ax$ terms to get:

$2x - 2ax - 7 - a^2 = b$

Factoring Out the Common Term

We can factor out the common term $2x$ from the first two terms on the left-hand side of the equation. This gives us:

$2x(1 - a) - 7 - a^2 = b$

Simplifying the Equation

Now that we have factored out the common term, we can simplify the equation further. We can combine the $-7$ and $-a^2$ terms to get:

$2x(1 - a) - (7 + a^2) = b$

Equating the Coefficients

Our goal is to find the values of $a$ and $b$ that satisfy the equation. To do this, we can equate the coefficients of the $x$ terms on both sides of the equation. The coefficient of the $x$ term on the left-hand side of the equation is $2(1 - a)$, and the coefficient of the $x$ term on the right-hand side of the equation is $0$. Equating these coefficients, we get:

$2(1 - a) = 0$

Solving for a

Now that we have equated the coefficients, we can solve for $a$. To do this, we can divide both sides of the equation by $2$ and then subtract $1$ from both sides. This gives us:

$1 - a = 0$

$a = 1$

Substituting the Value of a

Now that we have found the value of $a$, we can substitute it into the original equation. Substituting $a = 1$ into the equation, we get:

$x^2 + 2x - 7 = (x + 1)^2 + b$

Expanding the Right-Hand Side

We can expand the right-hand side of the equation by squaring the binomial $(x + 1)$. This gives us:

$x^2 + 2x - 7 = x^2 + 2x + 1 + b$

Simplifying the Equation

Now that we have expanded the right-hand side of the equation, we can simplify it by combining like terms. The $x^2$ terms on both sides of the equation cancel each other out, leaving us with:

$-7 = 1 + b$

Solving for b

Now that we have simplified the equation, we can solve for $b$. To do this, we can subtract $1$ from both sides of the equation. This gives us:

$-8 = b$

Conclusion

In this article, we have solved the quadratic equation $x^2 + 2x - 7 = (x + a)^2 + b$ to find the values of $a$ and $b$. We found that $a = 1$ and $b = -8$. This solution satisfies the original equation and provides a clear understanding of the relationship between the coefficients of the quadratic equation.

Final Answer

The final answer is:

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Introduction

In our previous article, we solved the quadratic equation $x^2 + 2x - 7 = (x + a)^2 + b$ to find the values of $a$ and $b$. In this article, we will answer some frequently asked questions related to the solution of this equation.

Q: What is the significance of the value of a in the quadratic equation?

A: The value of $a$ in the quadratic equation represents the coefficient of the $x$ term in the expanded form of $(x + a)^2$. In the solution of the equation, we found that $a = 1$. This value of $a$ allows us to rewrite the equation in a simpler form and solve for the value of $b$.

Q: How did you simplify the equation to find the value of b?

A: To simplify the equation, we expanded the right-hand side of the equation by squaring the binomial $(x + 1)$. This gave us $x^2 + 2x + 1 + b$. We then combined like terms and simplified the equation to get $-7 = 1 + b$. Finally, we solved for $b$ by subtracting $1$ from both sides of the equation.

Q: What is the relationship between the coefficients of the quadratic equation and the values of a and b?

A: The coefficients of the quadratic equation are related to the values of $a$ and $b$ through the expanded form of $(x + a)^2$. In the solution of the equation, we found that $a = 1$ and $b = -8$. These values of $a$ and $b$ satisfy the original equation and provide a clear understanding of the relationship between the coefficients of the quadratic equation.

Q: Can you provide a step-by-step solution to the quadratic equation?

A: Yes, we can provide a step-by-step solution to the quadratic equation. Here is the solution:

1. Expand the right-hand side of the equation by squaring the binomial $(x + a)$.
2. Combine like terms and simplify the equation.
3. Equate the coefficients of the $x$ terms on both sides of the equation.
4. Solve for $a$ by dividing both sides of the equation by $2$ and then subtracting $1$ from both sides.
5. Substitute the value of $a$ into the original equation and expand the right-hand side.
6. Simplify the equation by combining like terms and solving for $b$.

Q: What are some common mistakes to avoid when solving quadratic equations?

A: Some common mistakes to avoid when solving quadratic equations include:

• Not expanding the right-hand side of the equation correctly.
• Not combining like terms correctly.
• Not equating the coefficients of the $x$ terms on both sides of the equation.
• Not solving for $a$ and $b$ correctly.

Q: Can you provide some examples of quadratic equations that can be solved using this method?

A: Yes, here are some examples of quadratic equations that can be solved using this method:

• $x^2 + 3x - 2 = (x + a)^2 + b$
• $x^2 - 4x + 3 = (x + a)^2 + b$
• $x^2 + 2x - 5 = (x + a)^2 + b$

Conclusion

In this article, we have answered some frequently asked questions related to the solution of the quadratic equation $x^2 + 2x - 7 = (x + a)^2 + b$. We have provided a step-by-step solution to the equation and highlighted some common mistakes to avoid when solving quadratic equations. We have also provided some examples of quadratic equations that can be solved using this method.

Final Answer

The final answer is:

$a = 1$ $b = -8$