Question

Solve the equation: \[ \frac{(3x - 4)^2}{9} - \frac{(4x - 3)^2}{8} = 1 \quad \text{(length of the latus rectum)} \]

Hint:

When solving quadratic equations with negative discriminants, the solutions will be complex numbers.

Answer 1

We are given the equation: \[ \frac{(3x - 4)^2}{9} - \frac{(4x - 3)^2}{8} = 1 \] Step 1: Multiply the entire equation by the LCM of 9 and 8, which is 72.
Multiplying both sides by 72 to eliminate the denominators: \[ 72 \times \left( \frac{(3x - 4)^2}{9} - \frac{(4x - 3)^2}{8} \right) = 72 \times 1 \] Simplifying: \[ 8(3x - 4)^2 - 9(4x - 3)^2 = 72 \]
Step 2: Expand both squared terms.
First, expand \( (3x - 4)^2 \) and \( (4x - 3)^2 \): \[ (3x - 4)^2 = 9x^2 - 24x + 16 \] \[ (4x - 3)^2 = 16x^2 - 24x + 9 \] Now, substitute these expansions into the equation: \[ 8(9x^2 - 24x + 16) - 9(16x^2 - 24x + 9) = 72 \]
Step 3: Distribute the constants 8 and 9.
Distribute 8 and 9 to get: \[ 72x^2 - 192x + 128 - 144x^2 + 216x - 81 = 72 \]
Step 4: Combine like terms.
Now, combine like terms: \[ (72x^2 - 144x^2) + (-192x + 216x) + (128 - 81) = 72 \] \[ -72x^2 + 24x + 47 = 72 \]
Step 5: Simplify the equation.
Now, subtract 72 from both sides: \[ -72x^2 + 24x + 47 - 72 = 0 \] \[ -72x^2 + 24x - 25 = 0 \]
Step 6: Solve the quadratic equation.
Now, solve the quadratic equation \( -72x^2 + 24x - 25 = 0 \) using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] For the equation \( -72x^2 + 24x - 25 = 0 \), we have \( a = -72 \), \( b = 24 \), and \( c = -25 \). Substituting these values into the quadratic formula: \[ x = \frac{-24 \pm \sqrt{24^2 - 4(-72)(-25)}}{2(-72)} \] \[ x = \frac{-24 \pm \sqrt{576 - 7200}}{-144} \] \[ x = \frac{-24 \pm \sqrt{-6624}}{-144} \] Since the discriminant is negative, there are no real solutions. % Final Answer
Final Answer: \text{No real solutions.}