Discriminant Help

[12345678]

‘Calculate the discriminant of the quadratic 3x²+5x+8.
Hence show that 3x²+5x+8>0 for all values of x’.
So I understand that the discriminant is the b²-4ac from the quadratic formula. Using this, a=3, b=5 and c=8.
Therefore, the discriminant= 5²-(4*3*8)= 25-96= -71.
As the discriminant is a negative number, there are real roots for the equation, so how can the equation be greater than 0? Any help will be much appreciatedJ

 

[srmichael]

‘Calculate the discriminant of the quadratic 3x²+5x+8.
Hence show that 3x²+5x+8>0 for all values of x’.
So I understand that the discriminant is the b²-4ac from the quadratic formula. Using this, a=3, b=5 and c=8.
Therefore, the discriminant= 5²-(4*3*8)= 25-96= -71.
As the discriminant is a negative number, there are real roots for the equation (NO!), so how can the equation be greater than 0? Any help will be much appreciatedJ

If the discriminant is < 0, then there are no real roots. Instead, there are two imaginary roots.

Discriminant > 0: Two real roots (if the discriminant is a perfect square the roots are rational. If it is not a perfect square, they are irrational)

Discriminant = 0: One real root (technically, a root with a multiplicity of 2...also called a double root)

Discriminant < 0: Two imaginary roots

And what your problem is asking is that since you have two imaginary roots, the parabola is not crossing the x-axis and since the parabola is opening up (a > 0), then parabola is completely above the x -axis and thus > 0 for all x.

 

[12345678]

If the discriminant is < 0, then there are no real roots. Instead, there are two imaginary roots.

Discriminant > 0: Two real roots (if the discriminant is a perfect square the roots are rational. If it is not a perfect square, they are irrational)

Discriminant = 0: One real root (technically, a root with a multiplicity of 2...also called a double root)

Discriminant < 0: Two imaginary roots

And what your problem is asking is that since you have two imaginary roots, the parabola is not crossing the x-axis and since the parabola is opening up (a > 0), then parabola is completely above the x -axis and thus > 0 for all x.

So your essentially saying as the curve is above the x axis, it cannot intercept the x axis so cannot be 0. Also, what do you mean by the curve is opening up- does a curve not get narrower if a=1/2?

 

[srmichael]

So your essentially saying as the curve is above the x axis, it cannot intercept the x axis so cannot be 0. Also, what do you mean by the curve is opening up- does a curve not get narrower if a=1/2?

Since the discriminant is < 0, then the parabola is not crossing the x-axis, otherwise we would have two real roots (or one if it is touching the x-axis). And parabolas either open up or down. That is determined by the sign of the "a" coefficient on the x² term. Since this "a" = 3, the graph is opening up and since the discriminant is < 0, the parabola is entirely above the x-axis, meaning it is > 0 for all x.

Make sense?

 

[12345678]

Since the discriminant is < 0, then the parabola is not crossing the x-axis, otherwise we would have two real roots (or one if it is touching the x-axis). And parabolas either open up or down. That is determined by the sign of the "a" coefficient on the x² term. Since this "a" = 3, the graph is opening up and since the discriminant is < 0, the parabola is entirely above the x-axis, meaning it is > 0 for all x.

Make sense?

Yeah, I understand it now Thanks for your help, much appreciated!