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## 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 ax

^{2}+bx+c=0 (a≠0) have a real root is that b

^{2}-4ac>5.

**2**.A necessary condition that ax

^{2}+bx+c=0 (a≠0) have a real root is that b

^{2}-4ac=0.

## Homework Equations

∀

_{x}(p

_{x}→q

_{x})

## The Attempt at a Solution

1.[/B]A sufficient condition that ax

^{2}+bx+c=0 (a≠0) have a real root is that b

^{2}-4ac>5.

If b

^{2}-4ac>5, then ax

^{2}+bx+c=0 (a≠0) has a real root.

**ax**

^{2}+bx+c=0x

^{2}+(b/a)x+(c/a)=0

x

^{2}+(b/a)x+(b

^{2}/4a

^{2})+(c/a)=(b

^{2}/4a

^{2})

(x+(b/2a))

^{2}+(c/a)=(b

^{2}/4a

^{2})

(x+(b/2a))

^{2}=(b

^{2}/4a

^{2})-(c/a)

(x+(b/2a))

^{2}=(b

^{2}/4a

^{2})-(4ac/4a

^{2})

x+(b/2a)=(±√b

^{2}-4ac)/2a

**x=(-b±√b**

±√

^{2}-4ac)/2a±√

**b**^{2}-4ac≥0

**b**≥0^{2}-4ac**Since 5>0,**

2. A necessary condition that ax

**b**

^{2}-4ac>5 is sufficient condition for the equation to a have real root.2. A necessary condition that ax

^{2}+bx+c=0 (a≠0) have a real root is that b^{2}-4ac=0.If and only if b

^{2}-4ac=0, then ax

^{2}+bx+c=0 (a≠0) has a real root.

From

**1.**we know that ax

^{2}+bx+c=0 (a≠0) has a real root when

**.**

**b**^{2}-4ac>5Therefore it is false that the equation has a real root if and only if the discriminant is zero.