Multiple choice

Find the value of p for which the given equation has real roots. $\displaystyle 8p{ x }^{ 2 }-9x+3=0$

  1. $\displaystyle p\le \frac { 27 }{ 32 } $
  2. $\displaystyle p\ge \frac { 27 }{ 32 } $
  3. $\displaystyle p\le \frac { 32 }{ 27 } $
  4. $\displaystyle p\le \frac { 36 }{ 27 } $
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A Correct answer
Explanation

For real roots, the discriminant D >= 0. D = (-9)^2 - 4(8p)(3) = 81 - 96p. 81 - 96p >= 0, so 96p <= 81, p <= 81/96 = 27/32.

AI explanation

For the quadratic equation 8px^2 - 9x + 3 = 0 to have real roots, its discriminant must be greater than or equal to zero. The discriminant is (-9)^2 - 4(8p)(3), which simplifies to 81 - 96p. Setting the inequality 81 - 96p >= 0 and solving for p yields p <= 81/96. Simplifying the fraction by dividing the numerator and denominator by 3 gives the result p <= 27/32.

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