Reasoning: The equation
3x^2 + bx + 3 = 0 has one real solution. What must be true about b?
I think it is: b must be the same as
-4(a)(c).
What are your thoughts?
you are almost correct.
we need b^2-4ac=0, so b^2=4ac
In this case, we must have b^2=36, or b=±6
Well, b must be the same as -4(a)(c) only if the quadratic equation has two real solutions. Since you mentioned that the equation has one real solution, we need to look at a different relationship.
In a quadratic equation of the form ax^2 + bx + c = 0, if the discriminant (which is b^2 - 4ac) equals zero, then the equation has one real solution. Therefore, in this case, the value of b must satisfy the condition:
b^2 - 4ac = 0
Since the equation is 3x^2 + bx + 3 = 0, we know that a = 3 and c = 3. Substituting these values into the condition, we get:
b^2 - 4(3)(3) = 0
b^2 - 36 = 0
So, b^2 = 36. Taking the square root of both sides, we get:
b = ±6
Therefore, for the equation to have one real solution, the value of b must be either 6 or -6.
To determine the conditions for b in the equation 3x^2 + bx + 3 = 0 to have one real solution, we need to consider the discriminant. The discriminant of a quadratic equation ax^2 + bx + c = 0 is given by Δ = b^2 - 4ac.
If a quadratic equation has exactly one real solution, the discriminant must be equal to zero, Δ = 0. This means that b^2 - 4ac = 0.
In the given equation, a = 3 and c = 3. Hence, substituting these values into the discriminant equation:
b^2 - 4ac = 0
b^2 - 4(3)(3) = 0
b^2 - 36 = 0
b^2 = 36
b = ± √36
b = ± 6
Therefore, for the equation 3x^2 + bx + 3 = 0 to have one real solution, b must be equal to ±6, not necessarily -4(a)(c).
To determine what must be true about b for the equation 3x^2 + bx + 3 = 0 to have only one real solution, we need to consider the discriminant of the quadratic equation. The discriminant is calculated as follows:
Discriminant (D) = b^2 - 4ac
In this case, a = 3 and c = 3 since the equation is given as 3x^2 + bx + 3 = 0.
For a quadratic equation to have one real solution, the discriminant D should be equal to zero. This is because when the discriminant is zero, the quadratic equation simplifies to a single solution, indicating that there is only one real root.
So, setting D = 0, we can solve for b:
0 = b^2 - 4ac
0 = b^2 - 4(3)(3)
0 = b^2 - 36
b^2 = 36
b = ±√36
b = ±6
Therefore, for the equation 3x^2 + bx + 3 = 0 to have one real solution, b must be either 6 or -6, not necessarily equal to -4ac.