MHB What is the value of n in 8x^2+4x+n if the expression has real roots?

[mathdad]

If the expression 8x^2+4x+n has real roots, find the value of n.

Can someone get me started? I found this question online and find it interesting.

 

[MarkFL]

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?

 

[mathdad]

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?

Are you talking about the discriminant?

 

[MarkFL]

Are you talking about the discriminant?

Yes, the discriminant is what's under the radical in the quadratic formula, and it cannot be negative if the roots ore real. :D

 

[mathdad]

Yes, the discriminant is what's under the radical in the quadratic formula, and it cannot be negative if the roots ore real. :D

I plug a = 8, b = 4 and c = n into b^2 - 4ac and equate to 0, right?

 

[MarkFL]

I plug a = 8, b = 4 and c = n into b^2 - 4ac and equate to 0, right?

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.

 

[mathdad]

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.

I totally get it. Interesting question.