Given that x=1.5 is the only solution to the equation 4x^2 - ax + b = 0, find the values of a and b. Show all workings.

Answer: a = 12, b = 9

If x = 1.5 is the only solution to the equation 4x² - ax + b = 0 then

4x² - ax + b = 4 (x - 1.5) ² |use p² - 2pq + q² = (p - q) ²

4x² - ax + b = 4 (x² - 2x · 1.5 + 1.5²)

4x² - ax + b = 4 (x² - 3x + 2.25)

4x² - ax + b = 4x² - 12x + 9 |subtract 4x² from both sides

-ax + b = - 12x + 9

therefore

-a = - 12 ⇒ a = 12 and b = 9