Multiple choice

If equation $\displaystyle { x }^{ 2 }+8x+p=0$ has real and distinct roots then

  1. $\displaystyle p\ge 16$
  2. $\displaystyle p>16$
  3. $\displaystyle p<16$
  4. $\displaystyle p\le 16$
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C Correct answer
Explanation

For real and distinct roots, the discriminant D = b^2 - 4ac > 0. Here, 8^2 - 4*1*p > 0, so 64 - 4p > 0, which means 4p < 64, or p < 16.

AI explanation

The condition for a quadratic equation to have real and distinct roots is that its discriminant (D = b squared minus 4ac) must be greater than zero. Substituting the values from x squared plus 8x plus p equals 0 gives 8 squared minus 4 times 1 times p is greater than 0. Simplifying this yields 64 minus 4p is greater than 0, so 4p is less than 64. Dividing by 4 gives the result p is less than 16.

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