Multiple choice

Find the value of $p$ in the equation, where the roots are real, $\displaystyle 5{ x }^{ 2 }+3x-p=0$

  1. $\displaystyle p\le \frac { 20 }{ 9 } $
  2. $\displaystyle p\ge \frac { 9 }{ 20 } $
  3. $\displaystyle p\ge \frac { -9 }{ 20 } $

  4. None
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

For real roots, discriminant D = b^2 - 4ac >= 0. 3^2 - 4(5)(-p) >= 0. 9 + 20p >= 0. 20p >= -9, so p >= -9/20.

AI explanation

For the quadratic equation 5x^2 + 3x - p = 0 to have real roots, the discriminant must be greater than or equal to zero. The discriminant is D = b^2 - 4ac = 3^2 - 4(5)(-p) = 9 + 20p. Setting the inequality 9 + 20p >= 0 and solving for p gives 20p >= -9. Dividing by 20, we find that p is greater than or equal to -9/20.

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