https://file.aiquickdraw.com/mathbot/file/1726197455382orwpvn11.jpg

We are given the equation:

\[
x^2 + 7x - 2 = (x + a)^2 + b
\]

We need to find the values of \( a \) and \( b \).

### Step 1: Expand the right-hand side
The right-hand side of the equation can be expanded using the identity \( (x + a)^2 = x^2 + 2ax + a^2 \). Thus, the equation becomes:

\[
x^2 + 7x - 2 = x^2 + 2ax + a^2 + b
\]

### Step 2: Compare both sides of the equation
Now, compare the coefficients of \( x^2 \), \( x \), and the constant terms on both sides of the equation:

- Coefficient of \( x^2 \): Both sides have \( x^2 \), so no change here.
- Coefficient of \( x \): On the left-hand side, the coefficient is 7. On the right-hand side, it is \( 2a \). So, we get the equation:
  \[
  2a = 7 \quad \Rightarrow \quad a = \frac{7}{2}
  \]

- Constant term: On the left-hand side, the constant is \( -2 \), and on the right-hand side, it is \( a^2 + b \). So, we get:
  \[
  a^2 + b = -2
  \]

### Step 3: Solve for \( b \)
We already know \( a = \frac{7}{2} \). Substitute this into the equation \( a^2 + b = -2 \):

\[
\left( \frac{7}{2} \right)^2 + b = -2 \quad \Rightarrow \quad \frac{49}{4} + b = -2
\]

Now, solve for \( b \):

\[
b = -2 - \frac{49}{4} = \frac{-8}{4} - \frac{49}{4} = \frac{-57}{4}
\]

### Final Answer:
- \( a = \frac{7}{2} \)
- \( b = \frac{-57}{4} \)

Let me know if you'd like more details or have any questions!

Here are five related questions for further practice:

1. How would you solve a similar equation if the coefficients were different?
2. What happens if \( b \) is not a constant but another variable?
3. How can you complete the square for a quadratic expression?
4. What are the steps to solving an equation involving \( (x + a)^2 \)?
5. How does this process change if the equation involves cube terms instead of squares?

**Tip**: When expanding binomials like \( (x + a)^2 \), always remember to use the identity \( (x + a)^2 = x^2 + 2ax + a^2 \) to avoid mistakes.

Find the value of a and b in the equation x^2 + 7x - 2 = (x + a)^2 + b.

This problem involves solving for a and b in the quadratic equation x^2 + 7x - 2 = (x + a)^2 + b. The solution covers expanding binomials and comparing coefficients to find the values of a and b. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation