Q. Solve the Equation \( 25x^2 = 16 \) for \( x \)

Solution

1. Start with the given equation:
   
   \(25x^{2} = 16\).

   We need to isolate \(x\). The first algebraic operation is to remove the coefficient 25 that multiplies \(x^{2}\). This is done by dividing both sides of the equation by 25.

2. Divide both sides by 25:
   
   \(\dfrac{25x^{2}}{25} = \dfrac{16}{25}\).

   Simplify the left side, since \(\dfrac{25x^{2}}{25} = x^{2}\). Thus we obtain:
   
   \(x^{2} = \dfrac{16}{25}\).

3. Next, solve for \(x\) by taking the square root of both sides. Because squaring is not one-to-one on the real numbers, taking the square root yields two possible signs. So we write:
   
   \(x = \pm \sqrt{\dfrac{16}{25}}\).

   Here the symbol \(\pm\) means we consider both the positive and negative square roots.

4. Simplify the square root of a quotient by taking the square root of numerator and denominator separately:
   
   \(\sqrt{\dfrac{16}{25}} = \dfrac{\sqrt{16}}{\sqrt{25}} = \dfrac{4}{5}\).

   Therefore:
   
   \(x = \pm \dfrac{4}{5}\).

5. Finally, state the solutions explicitly:
   
   \(x = \dfrac{4}{5}\) or \(x = -\dfrac{4}{5}\).

6. Optional check: substitute each solution into the original equation.
   
   For \(x = \dfrac{4}{5}\): \(25\left(\dfrac{4}{5}\right)^{2} = 25\cdot\dfrac{16}{25} = 16\).
   
   For \(x = -\dfrac{4}{5}\): \(25\left(-\dfrac{4}{5}\right)^{2} = 25\cdot\dfrac{16}{25} = 16\).
   
   Both satisfy the original equation, confirming the solutions are correct.