Math Problem Statement

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

Solution

The equation given is:

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 (x+a)2(x + a)^2

Expanding (x+a)2(x + a)^2:

(x+a)2=x2+2ax+a2(x + a)^2 = x^2 + 2ax + a^2

So the equation becomes:

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

Step 2: Compare the terms

We can now compare the coefficients of both sides of the equation. On the left-hand side, we have:

x2+7x2x^2 + 7x - 2

On the right-hand side, after expanding:

x2+2ax+(a2+b)x^2 + 2ax + (a^2 + b)

Now, compare the coefficients of the corresponding powers of xx:

  1. Coefficient of x2x^2: On both sides, the coefficient of x2x^2 is 1, so this doesn't give us any new information.

  2. Coefficient of xx: The coefficient of xx on the left side is 7, and on the right side, it is 2a2a. Therefore, we have the equation:

    2a=7a=722a = 7 \quad \Rightarrow \quad a = \frac{7}{2}

  3. Constant term: The constant term on the left side is 2-2, and on the right side, it is a2+ba^2 + b. So, we have:

    a2+b=2a^2 + b = -2

Step 3: Solve for bb

Substitute a=72a = \frac{7}{2} into a2+b=2a^2 + b = -2:

(72)2+b=2\left( \frac{7}{2} \right)^2 + b = -2

494+b=2\frac{49}{4} + b = -2

b=2494b = -2 - \frac{49}{4}

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

Final Answer:

The values of aa and bb are:

a=72,b=574a = \frac{7}{2}, \quad b = \frac{-57}{4}


Would you like further explanation on any step or details about solving similar problems?

Here are five related questions to explore:

  1. How do you expand expressions like (x+a)2(x + a)^2?
  2. What is the method of comparing coefficients in equations?
  3. How do you solve for unknowns when dealing with quadratic equations?
  4. Can you explain the process of expanding binomials?
  5. How do constant terms influence the solution of an equation?

Tip: When solving equations by comparing coefficients, always align similar powers of xx to set up simpler equations for unknowns.

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Equations
Expanding Binomials

Formulas

(x + a)^2 = x^2 + 2ax + a^2

Theorems

Method of Comparing Coefficients

Suitable Grade Level

Grades 9-11