Math Problem Statement

25(m squared plus 1) Squared - 9 (n sqaured minus 5) squared

Solution

The expression you've provided is:

25(m2+1)29(n25)225 \left( m^2 + 1 \right)^2 - 9 \left( n^2 - 5 \right)^2

Let's simplify it step-by-step.

Step 1: Expand both squared terms

First, expand each of the squared binomials.

For the first part: (m2+1)2=m4+2m2+1\left( m^2 + 1 \right)^2 = m^4 + 2m^2 + 1 Now, multiply by 25: 25(m2+1)2=25(m4+2m2+1)=25m4+50m2+2525 \left( m^2 + 1 \right)^2 = 25 \left( m^4 + 2m^2 + 1 \right) = 25m^4 + 50m^2 + 25

For the second part: (n25)2=n410n2+25\left( n^2 - 5 \right)^2 = n^4 - 10n^2 + 25 Now, multiply by 9: 9(n25)2=9(n410n2+25)=9n490n2+2259 \left( n^2 - 5 \right)^2 = 9 \left( n^4 - 10n^2 + 25 \right) = 9n^4 - 90n^2 + 225

Step 2: Combine both parts

Now substitute these into the original expression: 25(m2+1)29(n25)2=(25m4+50m2+25)(9n490n2+225)25 \left( m^2 + 1 \right)^2 - 9 \left( n^2 - 5 \right)^2 = (25m^4 + 50m^2 + 25) - (9n^4 - 90n^2 + 225)

Simplify by distributing the negative sign: =25m4+50m2+259n4+90n2225= 25m^4 + 50m^2 + 25 - 9n^4 + 90n^2 - 225

Now, combine like terms: =25m49n4+50m2+90n2+25225= 25m^4 - 9n^4 + 50m^2 + 90n^2 + 25 - 225 =25m49n4+50m2+90n2200= 25m^4 - 9n^4 + 50m^2 + 90n^2 - 200

Final Simplified Expression:

25m49n4+50m2+90n220025m^4 - 9n^4 + 50m^2 + 90n^2 - 200

Would you like further clarification or have any specific questions on the steps? Here are some related questions to deepen your understanding:

  1. How do you factor expressions like this if needed?
  2. What would happen if we substituted specific values for mm and nn?
  3. How would the result change if the numbers 25 and 9 were swapped in the original expression?
  4. Can this expression be factored further, and if so, how?
  5. How do we handle higher powers of mm and nn in algebraic simplifications?

Tip: Always expand binomials before trying to combine or simplify terms. This helps avoid mistakes and makes the problem easier to manage.

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Math Problem Analysis

Mathematical Concepts

Algebra
Binomial Expansion
Polynomials

Formulas

(a + b)^2 = a^2 + 2ab + b^2
(a - b)^2 = a^2 - 2ab + b^2

Theorems

Difference of squares

Suitable Grade Level

Grades 9-11