The discriminant of a quadratic and real solutions

[Keen94]

Homework Statement

Given a series of mathematical statements, some of which are true and some of which are false. Prove or Disprove:
1. A sufficient condition that ax²+bx+c=0 (a≠0) have a real root is that b²-4ac>5.
2.A necessary condition that ax²+bx+c=0 (a≠0) have a real root is that b²-4ac=0.

Homework Equations

∀_x(p_x→q_x)

The Attempt at a Solution

1.[/B]A sufficient condition that ax²+bx+c=0 (a≠0) have a real root is that b²-4ac>5.

If b²-4ac>5, then ax²+bx+c=0 (a≠0) has a real root.
ax²+bx+c=0
x²+(b/a)x+(c/a)=0
x²+(b/a)x+(b²/4a²)+(c/a)=(b²/4a²)
(x+(b/2a))²+(c/a)=(b²/4a²)
(x+(b/2a))²=(b²/4a²)-(c/a)
(x+(b/2a))²=(b²/4a²)-(4ac/4a²)
x+(b/2a)=(±√b²-4ac)/2a
x=(-b±√b²-4ac)/2a
±√b²-4ac≥0
b²-4ac≥0
Since 5>0, b²-4ac>5 is sufficient condition for the equation to a have real root.

2. A necessary condition that ax²+bx+c=0 (a≠0) have a real root is that b²-4ac=0.

If and only if b²-4ac=0, then ax²+bx+c=0 (a≠0) has a real root.
From 1. we know that ax²+bx+c=0 (a≠0) has a real root when b²-4ac>5.
Therefore it is false that the equation has a real root if and only if the discriminant is zero.

 

[LCKurtz]

Homework Statement

Given a series of mathematical statements, some of which are true and some of which are false. Prove or Disprove:
1. A sufficient condition that ax²+bx+c=0 (a≠0) have a real root is that b²-4ac>5.
2.A necessary condition that ax²+bx+c=0 (a≠0) have a real root is that b²-4ac=0.

Homework Equations

∀_x(p_x→q_x)

The Attempt at a Solution

1.[/B]A sufficient condition that ax²+bx+c=0 (a≠0) have a real root is that b²-4ac>5.

If b²-4ac>5, then ax²+bx+c=0 (a≠0) has a real root.
ax²+bx+c=0
x²+(b/a)x+(c/a)=0
x²+(b/a)x+(b²/4a²)+(c/a)=(b²/4a²)
(x+(b/2a))²+(c/a)=(b²/4a²)
(x+(b/2a))²=(b²/4a²)-(c/a)
(x+(b/2a))²=(b²/4a²)-(4ac/4a²)
x+(b/2a)=(±√b²-4ac)/2a
x=(-b±√b²-4ac)/2a
±√b²-4ac≥0
b²-4ac≥0
Since 5>0, b²-4ac>5 is sufficient condition for the equation to a have real root.

2. A necessary condition that ax²+bx+c=0 (a≠0) have a real root is that b²-4ac=0.

If and only if b²-4ac=0, then ax²+bx+c=0 (a≠0) has a real root.
From 1. we know that ax²+bx+c=0 (a≠0) has a real root when b²-4ac>5.
Therefore it is false that the equation has a real root if and only if the discriminant is zero.

For part 2 you are only asked whether the discriminant being zero is necessary to have a real root, not whether it is also sufficient (which it is). So you should only be concerned with the "only if" statement. I would leave out the red words.

 

[Keen94]

Gotcha', thanks.