The discussion revolves around determining the value of n in the quadratic expression 8x² + 4x + n such that the expression has real roots. Participants explore the concept of the discriminant and its implications for the roots of the quadratic equation.
Participants generally agree on the importance of the discriminant for determining the nature of the roots of the quadratic. However, the specific value of n that satisfies the condition for real roots remains unresolved, as no calculations or conclusions are presented.
The discussion does not include specific calculations or the final value of n, leaving the mathematical steps and assumptions implicit.
MarkFL said:Consider the general quadratic:
$$ax^2+bx+c$$
If this quadratic is to have real roots, then the discriminant has to be non-negative, that is:
$$b^2-4ac\ge0$$
What do you get when you apply this rule to the quadratic you posted?
RTCNTC said:Are you talking about the discriminant?
MarkFL said:Yes, the discriminant is what's under the radical in the quadratic formula, and it cannot be negative if the roots ore real. :D
RTCNTC said:I plug a = 8, b = 4 and c = n into b^2 - 4ac and equate to 0, right?
MarkFL said:You want to set the discriminant greater than or equal to zero, since you don't want it to be negative. If the discriminant is equal to zero, then you will have a repeated root, or a root of multiplicity 2. If the discriminant is greater than zero, then you will have two distinct roots.
To put this in graphical terms, when the discriminant is zero, then the parabola will be tangent to the $x$-axis, that is, the parabola will only touch the $x$-axis at one point. If the discriminant is greater than zero, then the parabola will cross the $x$-axis at two points, and if the discriminant is negative then the parabola will not touch the $x$-axis at all.