Question

How do you use the discriminant to find all values of c for which the equation \[3{{x}^{2}}-4x+c=0\] has two real roots.

Answer 1

Hint: In this problem, we have to find the values of c using the discriminant. We know that we will have two real roots if discriminant is greater than zero i.e., \[{{b}^{2}}-4ac>0\]. We have to compare the general quadratic equation and the given equation to find the value of a, b, c, then we can substitute in the discriminant to find the value of c.

Complete step by step answer:
We know that the given equation to be solved using discriminant is,
\[3{{x}^{2}}-4x+c=0\]
We also know that for an equation \[a{{x}^{2}}+bx+c=0\], we have two real roots if the discriminant is greater than zero.
\[{{b}^{2}}-4ac>0\]….. (1)
Now we can compare the given quadratic equation and the above general equation, we get
a = 3, b = -4, c = c.
we can substitute the above values in discriminant (1), we get
\[\begin{align}
  & \Rightarrow {{\left( -4 \right)}^{2}}-4\left( 3 \right)\left( c \right)>0 \\
 & \Rightarrow 16-12c>0 \\
\end{align}\]
Now we can write the above inequality as,
\[\Rightarrow 12c<16\]
Now we can divide by 12 on both the sides, and cancel similar terms to get
\[\Rightarrow c<\dfrac{4}{3}\] or \[c\in \left( -\infty ,\dfrac{4}{3} \right)\]
Therefore, the answer is \[c<\dfrac{4}{3}\] or \[c\in \left( -\infty ,\dfrac{4}{3} \right)\] .

Note:
Students make mistakes in writing the correct inequality symbol for the correct discriminant which should be concentrated. We should know that we will have two real roots if the discriminant is greater than zero i.e., \[{{b}^{2}}-4ac>0\]. We should also know how to deal with inequalities to solve these types of problems.