SOLUTION: this is a question i am stumped by.
THE SET OF ALL VALUES OF 'B' for which the equation 4x^2 +bx+1=0 has either one real root or two real roots is defined by what.
it says
Question 227633: this is a question i am stumped by.
THE SET OF ALL VALUES OF 'B' for which the equation 4x^2 +bx+1=0 has either one real root or two real roots is defined by what.
it says the answer is ('b' is greater to or equal to 4 (b>_4) or 'b' is less than or equal to 4 (b<_4)
HOW DOES THAT WORK. Answer by jsmallt9(3758) (Show Source):
You can put this solution on YOUR website! The general form for quadratic equations is: . And we have the quadratic formula for this form: .
A key in the formula is the expression in the square root: . It is such a key value it has been given a name: discriminant. Let's think about the possible values for the discriminant:
Positive: If the discriminant is positive the square root will be positive, too. And then there will be two real roots: One for and one for
Zero: If the discriminant is zero then the square root will be zero. And if the square root is zero then there will be only one real root. This is so because whether you add or subtract zero in the numerator, it works out the same.
Negative: If the discriminant is negative then there are no real roots because there are no real numbers you can square and end up with a negative result.
In your problem you want one or two real roots. So you want the discriminant to be positive or zero. IOW:
In your equation "a" is 4 and "c" is 1 and we're looking for "b":
Simplified we get:
We can solve this by factoring and then using our knowledge of multiplication. This is a difference of squares so it factors into:
Let's look at and think about what we have and see if you understand:
When two numbers are multiplied and the result is positive, then the two numbers are either both positive or both negative.
If a product is zero, one (or more) of the factors is zero.
No matter what number "b" is b+4 will always be larger than b-4.
If the smaller of two numbers is positive, then the larger number would have to be positive.
If the larger of two numbers is negative, then the smaller number is also negative.
says that a product is zero or positive. So we want the factors to be both positive or both negative. For both positive we just have to make sure the smaller factor is positive:
For both negative we just have to make sure the larger factor is negative:
(Note: The "or equal to" part of these inequalities handle the possible zero products.) So our solution is or
Solving each we get: or
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