When is a quadratic equation always positive?
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The quadratic formula and the discriminant
The quadratic formula was covered in the module Algebra review. This formula gives solutions to the general quadratic equation ax2+bx+c=0ax2+bx+c=0, when they exist, in terms of the coefficients a,b,ca,b,c. The solutions are
x=−b+b2−4ac−−−−−−−√2a,x=−b−b2−4ac−−−−−−−√2a,x=−b+b2−4ac2a,x=−b−b2−4ac2a,
provided that b2−4ac≥0b2−4ac≥0.
The quantity b2−4acb2−4ac is called the discriminant of the quadratic, often denoted by ΔΔ, and should be found first whenever the formula is being applied. It discriminates between the types of solutions of the equation:
Δ>0Δ>0 tells us the equation has two distinct real rootsΔ=0Δ=0 tells us the equation has one (repeated) real rootΔ<0Δ<0 tells us the equation has no real roots.
Screencast of interactive 2 , Interactive 2 
Exercise 4
For what values of kk does the equation (4k+1)x2−6kx+4=0(4k+1)x2−6kx+4=0 have one real solution?
A quadratic expression which always takes positive values is called positive definite, while one which always takes negative values is called negative definite.
Quadratics of either type never take the value 0, and so their discriminant is negative. Furthermore, such a quadratic is positive definite if a>0a>0, and negative definite if a<0a<0.

Detailed description of diagram
Example
Show that the quadratic expression 4x2−8x+74x2−8x+7 always takes positive values for any value of xx.
Solution
In this case, a=4a=4, b=−8b=−8 and c=7c=7. So
Δ=(−8)2−4×4×7=−48<0Δ=(−8)2−4×4×7=−48<0
and a=4>0a=4>0. Hence the quadratic is positive definite.
The quadratic formula was covered in the module Algebra review. This formula gives solutions to the general quadratic equation ax2+bx+c=0ax2+bx+c=0, when they exist, in terms of the coefficients a,b,ca,b,c. The solutions are
x=−b+b2−4ac−−−−−−−√2a,x=−b−b2−4ac−−−−−−−√2a,x=−b+b2−4ac2a,x=−b−b2−4ac2a,
provided that b2−4ac≥0b2−4ac≥0.
The quantity b2−4acb2−4ac is called the discriminant of the quadratic, often denoted by ΔΔ, and should be found first whenever the formula is being applied. It discriminates between the types of solutions of the equation:
Δ>0Δ>0 tells us the equation has two distinct real rootsΔ=0Δ=0 tells us the equation has one (repeated) real rootΔ<0Δ<0 tells us the equation has no real roots.
Screencast of interactive 2 , Interactive 2 
Exercise 4
For what values of kk does the equation (4k+1)x2−6kx+4=0(4k+1)x2−6kx+4=0 have one real solution?
A quadratic expression which always takes positive values is called positive definite, while one which always takes negative values is called negative definite.
Quadratics of either type never take the value 0, and so their discriminant is negative. Furthermore, such a quadratic is positive definite if a>0a>0, and negative definite if a<0a<0.

Detailed description of diagram
Example
Show that the quadratic expression 4x2−8x+74x2−8x+7 always takes positive values for any value of xx.
Solution
In this case, a=4a=4, b=−8b=−8 and c=7c=7. So
Δ=(−8)2−4×4×7=−48<0Δ=(−8)2−4×4×7=−48<0
and a=4>0a=4>0. Hence the quadratic is positive definite.
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