(x+2)2/5+(x-2)2/3=16/3

3 min read Jun 16, 2024

Solving the Equation: (x+2)²/5 + (x-2)²/3 = 16/3

This article will guide you through the steps of solving the equation (x+2)²/5 + (x-2)²/3 = 16/3.

Step 1: Find a Common Denominator

To combine the fractions on the left side of the equation, we need a common denominator. The least common multiple of 5 and 3 is 15. So, we will multiply the first term by 3/3 and the second term by 5/5:

(3/3) * (x+2)²/5 + (5/5) * (x-2)²/3 = 16/3

This gives us:

3(x+2)²/15 + 5(x-2)²/15 = 16/3

Step 2: Combine the Fractions

Now that the fractions have the same denominator, we can combine them:

[3(x+2)² + 5(x-2)²] / 15 = 16/3

Step 3: Simplify and Expand

Expand the squares in the numerator:

[3(x² + 4x + 4) + 5(x² - 4x + 4)] / 15 = 16/3

Simplify:

[3x² + 12x + 12 + 5x² - 20x + 20] / 15 = 16/3

Combine like terms:

[8x² - 8x + 32] / 15 = 16/3

Step 4: Solve for x

Multiply both sides of the equation by 15 to eliminate the denominator on the left side:

8x² - 8x + 32 = 80

Subtract 80 from both sides:

8x² - 8x - 48 = 0

Divide both sides by 8:

x² - x - 6 = 0

Factor the quadratic equation:

(x - 3)(x + 2) = 0

Therefore, the solutions are:

x = 3 or x = -2

Conclusion

By following these steps, we have successfully solved the equation (x+2)²/5 + (x-2)²/3 = 16/3. The solutions are x = 3 and x = -2.