To simplify the given algebraic expression, we will expand the squared terms and combine like terms.
Algebraic Expression Simplification
The expression to simplify is: $(5z-12y)^2 + (12z+5y)^2 - 144z^2$.
Expanding $(5z-12y)^2$
Use the algebraic identity $(a-b)^2 = a^2 - 2ab + b^2$. Here, $a = 5z$ and $b = 12y$.
$ (5z-12y)^2 = (5z)^2 - 2(5z)(12y) + (12y)^2 $ $ = 25z^2 - 120zy + 144y^2 $Expanding $(12z+5y)^2$
Use the algebraic identity $(a+b)^2 = a^2 + 2ab + b^2$. Here, $a = 12z$ and $b = 5y$.
$ (12z+5y)^2 = (12z)^2 + 2(12z)(5y) + (5y)^2 $ $ = 144z^2 + 120zy + 25y^2 $Combining Expanded Terms
Substitute the expanded forms back into the original expression:
$ (25z^2 - 120zy + 144y^2) + (144z^2 + 120zy + 25y^2) - 144z^2 $Simplifying Like Terms
Group the terms involving $z^2$, $zy$, and $y^2$ separately.
- $z^2$ terms: $25z^2 + 144z^2 - 144z^2 = 25z^2$
- $zy$ terms: $-120zy + 120zy = 0$
- $y^2$ terms: $144y^2 + 25y^2 = 169y^2$
Final Simplified Expression
Combine the simplified terms:
$ 25z^2 + 0 + 169y^2 = 25z^2 + 169y^2 $The simplified expression is $25z^2 + 169y^2$.
Important Questions from Simplification (Notes)
The value of $(-27) \times (-16) + (-27) \times (-14)$ is
- Evaluate: (-9) - (-56) ÷ (-14) + (-4) × 7
- $\frac{\sqrt[3]{6859}}{\sqrt[4]{256}} \times \frac{2}{57} \times 168 = ?$
- In the Delhi zoo, there are some ducks and rabbits. If the heads are counted there are 160, while the legs are 450. What will be number of ducks in the zoo ?
Identify the number that will replace the question mark in the second equation based on the relationship represented in the first equation.
