Let $3x^2 + 4x + c = 0.$ What value of $c$ will give us exactly one solution for $x$?
3x^2 + 4x + c = 0 , ...... (skip all that $ stuff when typing it here)
You have one real root when the discriminant is equal to zero
b^2 - 4ac = 0
4^2 - 4(3)(c) = 0
12c = 16
c = 16/12 = 4/3
Well, I can tell you that $c$ stands for "chuckle." So if we want exactly one solution for $x$, we need to solve for $c = \text{chuckle}$. But I must warn you, solving for "chuckle" might result in a lot of laughter!
To find the value of $c$ that will give us exactly one solution for $x$, we need to consider the discriminant of the quadratic equation.
The discriminant is given by the formula $\Delta = b^2 - 4ac$, where $a$, $b$, and $c$ are the coefficients of the quadratic equation $ax^2 + bx + c = 0$.
For our equation $3x^2 + 4x + c = 0$, we have $a = 3$, $b = 4$, and $c$ is the value we are trying to find.
If the discriminant $\Delta$ is equal to $0$, then the quadratic equation has exactly one solution. So we need to find the value of $c$ that makes $\Delta$ equal to $0$.
We can substitute the values of $a$, $b$, and $c$ into the discriminant formula: $\Delta = (4)^2 - 4(3)(c)$.
Simplifying, we have $\Delta = 16 - 12c$.
Now, we set $\Delta$ equal to $0$: $16 - 12c = 0$.
Solving for $c$, we get $c = \frac{16}{12} = \frac{4}{3}$.
Therefore, the value of $c$ that will give us exactly one solution for $x$ is $c = \frac{4}{3}$.
To find the value of $c$ that will give us exactly one solution for $x$, we need to apply the quadratic formula, which states that the solutions to a quadratic equation of the form $ax^2 + bx + c = 0$ are given by:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
In order for the equation to have exactly one solution, the discriminant, $b^2 - 4ac$, must be equal to zero. Specifically for our equation $3x^2 + 4x + c = 0$, this condition can be expressed as:
\[4^2 - 4(3)(c) = 0\]
Simplifying this equation, we get:
\[16 - 12c = 0\]
To solve for $c$, we can isolate it by moving the $16$ to the other side:
\[12c = 16\]
Finally, we can solve for $c$ by dividing both sides of the equation by $12$:
\[c = \frac{16}{12} = \frac{4}{3}\]
Therefore, the value of $c$ that will give us exactly one solution for $x$ is $\frac{4}{3}$.