If `7/3 x^(2) - 10 = 7/5 x^(2) + 200` , then what can be the value of x ?

Options

A. 5
B. 10
C. 15
D. 20

Answer

To solve the equation \( \frac{7}{3} x^2 - 10 = \frac{7}{5} x^2 + 200 \), we will follow these steps:

### Step 1: Rearranging the equation
We want to move all terms involving \( x^2 \) to one side and the constant terms to the other side. We can do this by subtracting \( \frac{7}{5} x^2 \) from both sides and adding 10 to both sides.

\[ \frac{7}{3} x^2 - \frac{7}{5} x^2 = 200 + 10 \]

### Step 2: Simplifying the equation
Now, simplify the right side:

\[ \frac{7}{3} x^2 - \frac{7}{5} x^2 = 210 \]

### Step 3: Finding a common denominator
To combine the terms on the left side, we need a common denominator. The least common multiple of 3 and 5 is 15.

Convert each fraction:

\[ \frac{7}{3} = \frac{35}{15} \quad \text{and} \quad \frac{7}{5} = \frac{21}{15} \]

Now substitute these back into the equation:

\[ \frac{35}{15} x^2 - \frac{21}{15} x^2 = 210 \]

### Step 4: Combining the fractions
Now, combine the fractions on the left:

\[ \frac{35 - 21}{15} x^2 = 210 \]

This simplifies to:

\[ \frac{14}{15} x^2 = 210 \]

### Step 5: Isolating \( x^2 \)
To isolate \( x^2 \), multiply both sides by the reciprocal of \( \frac{14}{15} \):

\[ x^2 = 210 \times \frac{15}{14} \]

### Step 6: Simplifying the right side
Now calculate \( 210 \times \frac{15}{14} \):

First, simplify \( \frac{210}{14} = 15 \):

\[ x^2 = 15 \times 15 = 225 \]

### Step 7: Taking the square root
Finally, take the square root of both sides to find \( x \):

\[ x = \sqrt{225} = 15 \]

### Conclusion
Thus, the value of \( x \) is \( 15 \).

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