Answer:
For values of p less than or equal to 9/16, the equation x(4x+3)=-p will have real roots.
Explanation:
The given equation is x(4x+3)=-p.
To find the values of p for which the equation has real roots, we can use the discriminant of the quadratic equation.
The quadratic equation formed by the given equation is 4x^2 + 3x + p = 0.
The discriminant of this quadratic equation is:
D = b^2 - 4ac
= 3^2 - 4(4)(p)
= 9 - 16p
For the equation to have real roots, the discriminant must be non-negative, i.e., D ≥ 0.
Therefore, we have:
9 - 16p ≥ 0
Solving for p, we get:
p ≤ 9/16
Hence, for values of p less than or equal to 9/16, the equation x(4x+3)=-p will have real roots. For values of p greater than 9/16, the equation will have complex roots.