How do you solve a quadratics equation without the value of b being given to find 2 irrational solutions?

Question

i've been given the equation 4x^2+bx+9 and i must find a value for b that will provided me with 2 irrational solutions...HELP PLEASE!!!

Answer 1

Remember that all you really need to do is to calculate the discriminant, b^2 - 4ac, where a, b, c are the coefficients of your quadratic equation. Then you can use the following cases to determine the types of roots you will get.

b^2 - 4ac > 0 AND b^2 - 4ac is a perfect square
Solution: two rational roots

b^2 - 4ac > 0 AND b^2 - 4ac is not a perfect square
Solution: two irrational roots

b^2 - 4ac = 0
Solution: one rational root

b^2 - 4ac < 0
Solution: two imaginary roots (conjugates of each other)

In your case 4x^2 + bx + 9 gives:
a = 4
b = b
c = 9

So the discriminant becomes:
b^2 - 4ac = b^2 - 4(4)(9) = b^2 - 144

To get two irrational roots, you need b^2 - 144 > 0 AND b^2 - 144 to not be a perfect square.

b^2 - 144 > 0
b^2 > 144
b > 12 OR b < -12

Suppose b = 13.
Then (13)^2 - 144 = 169 - 144 = 25
Now, 25 > 0 BUT 25 is a perfect square and this is not what you want.

Suppose b = 14.
(14)^2 - 144 = 196 - 144 = 52
52 > 0 and 52 is not a perfect square (i.e. nothing "squared" gives 52).

So a good answer would be b = 14.

Answer 2

since x = (-b ± √(b² - 4ac))/2a, the key is the √(b²-4ac) term. If it is irrational, then both answers will be irrational.

Sooo......... 4ac = 4*4*9 = 144 so you want a b value such that b² > 144 (or your answers will be complex as well as irrational ☺) and you need for b² - 4ac to not be equal to a perfect square. I'm guessing that your instructor is expecting an integer value for b, so check
12² = 144, 13² = 169 and
169 - 144 = 25 = 5² so using b = 13 won't work. But
14² = 196 and 196 - 144 = 52 and √52 = 2√13
and √13 is irrational, so use 14 for your b value.

Hope that helps ☺


Doug

Answer 3

4 x^2 + bx + 9 can be expressed as (2x + 3)^2 and if we expand (2x+3)^2 we get 4x^2 + 12x + 9 thus giving x = -3/2

We can also get the same 4x^2 + bx + 9 by squaring 2x - 3, which gives us 4x^2 - 12x + 9 and that gives us x = +3/2

Both of the solutions are valid since b is not defined.

Answer 4

2 is the "a" time era interior the quadratic formulation: roots are (-b +/- ?(b^2 - 4ax) ) / 2a So purely follow the quadratic formulation interior the customary way. in case you're attempting to ingredient it, undergo in ideas the rational roots attempt. Any root a/b with a and b in lowest words ought to have a dividing 5 (so it relatively is one million or 5), and b dividing 2 (so it relatively is one million or 2). so which you have have been given something like (2x +/- 5) or (2x +/- one million) as a ingredient. BTW, +/- ability plus or minus, so i did no longer ought to jot down out the two words.)