Determine the value of "m" that will give the quadratic 3x^2 + 4x + m = 0 :
a) two equal roots
b) no real roots.
For two equal roots, the discriminant has to be zero, ...
b^2 - 4ac = 0
16 - 4(3)(m) = 0
16 = 12m
m = ....
For no real roots, b^2 - 4ac < 0 , so ....
To determine the value of "m" that will give the quadratic 3x^2 + 4x + m = 0 a) two equal roots and b) no real roots, we can use the discriminant.
The discriminant, denoted as "Δ," is a value that helps determine the nature of the roots of a quadratic equation. For a quadratic equation in the form ax^2 + bx + c = 0, the discriminant is given by the formula: Δ = b^2 - 4ac.
a) For two equal roots, the discriminant should be equal to zero (Δ = 0).
Δ = (4)^2 - 4(3)(m)
Δ = 16 - 12m
To have two equal roots, we want Δ = 0:
16 - 12m = 0
Solving this equation for "m," we have:
16 = 12m
m = 16/12
m = 4/3
Therefore, to have two equal roots, the value of "m" should be 4/3.
b) For no real roots, the discriminant should be less than zero (Δ < 0).
Δ = (4)^2 - 4(3)(m)
Δ = 16 - 12m
To have no real roots, we want Δ < 0:
16 - 12m < 0
Solving this inequality for "m," we have:
-12m < -16
m > -16/-12 (since we flipped the inequality due to multiplying by negative)
m > 4/3
Therefore, to have no real roots, the value of "m" should be greater than 4/3.
To determine the value of "m" that will give the quadratic equation 3x^2 + 4x + m = 0:
a) For the quadratic equation to have two equal roots, the discriminant (b^2 - 4ac) must be equal to zero.
Let's calculate the discriminant for our equation:
Discriminant = 4^2 - 4*3*m
Since we want the discriminant to be zero, we set it equal to zero and solve:
16 - 12m = 0
Rearranging the equation, we have:
12m = 16
Finally, we solve for "m":
m = 16/12
m = 4/3
Therefore, for the quadratic equation to have two equal roots, the value of "m" should be 4/3.
b) For the quadratic equation to have no real roots, the discriminant must be negative.
We calculate the discriminant for our equation again:
Discriminant = 4^2 - 4*3*m
For no real roots, the discriminant should be negative:
16 - 12m < 0
Now, we solve this inequality:
-12m < -16
Dividing both sides by -12 (and reversing the inequality sign since we are dividing by a negative number):
m > 16/12
m > 4/3
Therefore, for the quadratic equation to have no real roots, the value of "m" should be greater than 4/3.