Simplification Problems

The following expressions require expansion using algebraic identities and further simplification:

1. Simplify: $(4x – 3y)^2 + 48xy$

2. Simplify: $(7a + 2b)^2 – 56ab$

3. Simplify: $(x – \frac{3}{x})^2 + 12$

4. Simplify: $(p + \frac{4}{p})^2 – 16$

5. Simplify: $(5x + 4y)^2 – 40xy$

6. Simplify: $(ab + 2c)^2 – 4abc$

7. Simplify: $(m^2 – 3n^2)^2 + 6m^2n^2$

Proofs and Expressions

These problems involve proving equalities or rewriting terms using specific identities:

3. Show that: $(6m – n)^2 + 24mn = (6m + n)^2$

4. Express in terms of squares: $9x^2 + 16y^2$ (using suitable identity)

5. Express in terms of squares: $25a^2 + b^2$ (using suitable identity)

To solve these problems, we primarily use the square of a binomial identities:

1. $(a + b)^2 = a^2 + 2ab + b^2$

2. $(a – b)^2 = a^2 – 2ab + b^2$

Here are the step-by-step explanations for each problem:

1.Simplify: $(4x – 3y)^2 + 48xy$

• Expand: Use $(a – b)^2$. Here, $a = 4x$ and $b = 3y$. This gives $(4x)^2 – 2(4x)(3y) + (3y)^2$, which is $16x^2 – 24xy + 9y^2$.

• Simplify: Add the $48xy$ term: $16x^2 – 24xy + 9y^2 + 48xy$.

• Result: Combine like terms ($-24xy + 48xy = +24xy$) to get $16x^2 + 24xy + 9y^2$, which can also be written as $(4x + 3y)^2$.

2.Simplify: $(7a + 2b)^2 – 56ab$

• Expand: Use $(a + b)^2$. This results in $49a^2 + 28ab + 4b^2$.

• Simplify: Subtract $56ab$ from the expansion.

• Result: $49a^2 – 28ab + 4b^2$, which simplifies back to $(7a – 2b)^2$.

3.Show that: $(6m – n)^2 + 24mn = (6m + n)^2$

• Left Hand Side (LHS): Expanding $(6m – n)^2$ gives $36m^2 – 12mn + n^2$.

• Add $24mn$: $36m^2 – 12mn + n^2 + 24mn$ simplifies to $36m^2 + 12mn + n^2$.

• Right Hand Side (RHS): Expanding $(6m + n)^2$ also gives $36m^2 + 12mn + n^2$. Since LHS = RHS, the identity is proved.

4.Simplify: $(x – \frac{3}{x})^2 + 12$

• Expand: The middle term of the expansion is $-2(x)(\frac{3}{x})$, which equals $-6$. So, the expansion is $x^2 – 6 + \frac{9}{x^2}$.

• Simplify: $x^2 – 6 + \frac{9}{x^2} + 12$.

• Result: $x^2 + 6 + \frac{9}{x^2}$, which is the square of $(x + \frac{3}{x})$.

5.Simplify: $(p + \frac{4}{p})^2 – 16$

• Expand: The middle term is $2(p)(\frac{4}{p}) = 8$. The expansion is $p^2 + 8 + \frac{16}{p^2}$.

• Simplify: Subtract 16: $p^2 + 8 + \frac{16}{p^2} – 16$.

• Result: $p^2 – 8 + \frac{16}{p^2}$, which simplifies to $(p – \frac{4}{p})^2$.

6.Simplify: $(5x + 4y)^2 – 40xy$

• Expand: $(5x + 4y)^2 = 25x^2 + 40xy + 16y^2$.

• Simplify: Subtracting $40xy$ cancels out the middle term

• Result: $25x^2 + 16y^2$.

7.Simplify: $(ab + 2c)^2 – 4abc$

• Expand: $(ab)^2 + 2(ab)(2c) + (2c)^2 = a^2b^2 + 4abc + 4c^2$.

• Simplify: Subtract $4abc$ from the expansion.

• Result: $a^2b^2 + 4c^2$.

8.Simplify: $(m^2 – 3n^2)^2 + 6m^2n^2$

• Expand: $(m^2)^2 – 2(m^2)(3n^2) + (3n^2)^2 = m^4 – 6m^2n^2 + 9n^4$.

• Simplify: Add $6m^2n^2$ to the expansion.

• Result: $m^4 + 9n^4$.

9.Express in terms of squares: $9x^2 + 16y^2$

• Identity: Use the variation $a^2 + b^2 = (a + b)^2 – 2ab$.

• Application: Here $a = 3x$ and $b = 4y$.

• Result: $(3x + 4y)^2 – 2(3x)(4y) = (3x + 4y)^2 – 24xy$.

10.Express in terms of squares: $25a^2 + b^2$

• Identity: Similar to problem 9, use $a^2 + b^2 = (a – b)^2 + 2ab$.

• Application: Here $a = 5a$ and $b = b$.

• Result: $(5a – b)^2 + 10ab$ or $(5a + b)^2 – 10ab$.