Math Problem Statement

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

Solution

We are given the equation:

x2+7x2=(x+a)2+bx^2 + 7x - 2 = (x + a)^2 + b

We need to find the values of aa and bb.

Step 1: Expand the right-hand side

The right-hand side of the equation can be expanded using the identity (x+a)2=x2+2ax+a2(x + a)^2 = x^2 + 2ax + a^2. Thus, the equation becomes:

x2+7x2=x2+2ax+a2+bx^2 + 7x - 2 = x^2 + 2ax + a^2 + b

Step 2: Compare both sides of the equation

Now, compare the coefficients of x2x^2, xx, and the constant terms on both sides of the equation:

  • Coefficient of x2x^2: Both sides have x2x^2, so no change here.

  • Coefficient of xx: On the left-hand side, the coefficient is 7. On the right-hand side, it is 2a2a. So, we get the equation: 2a=7a=722a = 7 \quad \Rightarrow \quad a = \frac{7}{2}

  • Constant term: On the left-hand side, the constant is 2-2, and on the right-hand side, it is a2+ba^2 + b. So, we get: a2+b=2a^2 + b = -2

Step 3: Solve for bb

We already know a=72a = \frac{7}{2}. Substitute this into the equation a2+b=2a^2 + b = -2:

(72)2+b=2494+b=2\left( \frac{7}{2} \right)^2 + b = -2 \quad \Rightarrow \quad \frac{49}{4} + b = -2

Now, solve for bb:

b=2494=84494=574b = -2 - \frac{49}{4} = \frac{-8}{4} - \frac{49}{4} = \frac{-57}{4}

Final Answer:

  • a=72a = \frac{7}{2}
  • b=574b = \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 bb 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(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(x + a)^2, always remember to use the identity (x+a)2=x2+2ax+a2(x + a)^2 = x^2 + 2ax + a^2 to avoid mistakes.

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Equations
Binomial Expansion
Equation Comparison

Formulas

(x + a)^2 = x^2 + 2ax + a^2
Quadratic Expansion Formula

Theorems

Binomial Theorem
Quadratic Coefficient Matching

Suitable Grade Level

Grades 9-11