SOLUTION: decide all values of b in the following equations that will give one or more real number solutions. a. 3x^2+bx-3=0 b. 5x^2+bx+1=0 c. -3x^2+bx-3=0 d. Write a rule for judging

Question 91699: decide all values of b in the following equations that will give one or more real number solutions.

a. 3x^2+bx-3=0
b. 5x^2+bx+1=0
c. -3x^2+bx-3=0
d. Write a rule for judging if an equation has solutions by looking at it in standard form.
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
a)
Remember, real solutions result when the discriminant is greater than or equal to zero.

So we can use this:

note: remember, in the quadratic formula , the discriminant consists of every term in the square root. Since you cannot take the square root of a negative number, the discriminant must be positive.

Now looking at we see that a=3 and c=-3

Plug in a=3 and c=-3

Multiply

Subtract 36 from both sides

Since is always positive, is always true. So b can be any real number

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b)
Again, using the discriminant we get:

Now looking at we see that a=5 and c=1

Plug in a=5 and c=1

Multiply

Add 20 to both sides

or Take the square root of both sides

So b can be greater than or equal to or less than or equal to (but not both at the same time)

------------------------------------------------------------------------

Again, using the discriminant we get:

Now looking at we see that a=-3 and c=-3

Plug in a=-3 and c=-3

Multiply

Subtract 36 from both sides

or Take the square root of both sides

So b can be greater than 6 or less than -6 (but not both at the same time)

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d)
The general rule (the one we've been using) is the discriminant must be greater than or equal to zero in order to produce real solutions. Remember, you cannot take the square root of a negative number, so the discriminant must be positive.

So the general rule is

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