If $8a=35^{2}-27^{2}$ then find a

Question

Accepted Answer

The correct option is 1:
To solve this problem, we need to find the value of 'a' from the given equation $8a=35^{2}-27^{2}$. We can simplify the right-hand side of the equation using a standard algebraic identity.

Key Concept: Difference of Squares Identity
The difference of squares identity states that for any two numbers $x$ and $y$:
$x^2 - y^2 = (x + y)(x - y)$

Step-by-step Explanation:1.  Identify the values for $x$ and $y$ in the given equation:
    In $35^2 - 27^2$, we have $x = 35$ and $y = 27$.

2.  Apply the difference of squares identity:
    Substitute the values of $x$ and $y$ into the identity:
    $35^2 - 27^2 = (35 + 27)(35 - 27)$

3.  Perform the addition and subtraction within the parentheses:
    $35 + 27 = 62$
    $35 - 27 = 8$

4.  Substitute these results back into the equation:
    $35^2 - 27^2 = (62)(8)$
    $35^2 - 27^2 = 496$

5.  Now, substitute this value back into the original equation:
    $8a = 496$

6.  Solve for 'a' by dividing both sides by 8:
    $a = \frac{496}{8}$
    $a = 62$Therefore, the value of 'a' is 62, which corresponds to Option 1.

The incorrect option is 2:
Option 2 (64) is incorrect. This might result from an arithmetic error during addition or subtraction, or an incorrect application of the algebraic identity. For example, if someone incorrectly calculated $(35+27)$ or $(35-27)$.

The incorrect option is 3:
Option 3 (8) is incorrect. This value is obtained from $35-27$, which is only one part of the expanded difference of squares. It suggests that the calculation was incomplete or there was a misunderstanding of how to solve for 'a'.

The incorrect option is 4:
Option 4 (16) is incorrect. This value is not derived from any logical step in solving the equation. It might be a result of a miscalculation or an assumption not based on the given mathematical expression.