Question

What are the solutions of $3{{x}^{2}}+14x+8=0$

Answer 1

In the given question we are given a quadratic equation which can be solved by various methods like after making a factor and then equating each factor to zero and hence finding the values or by discriminant method.

Complete step by step solution:
The given equation is a quadratic equation of the form $a{{x}^{2}}+bx+c=0$
Here a=3, b=14, c=8
Now we know that the discriminant is given by $D=\sqrt{{{b}^{2}}-4ac}$
Therefore,
$\begin{aligned} & D=\sqrt{{{14}^{2}}-4\times 3\times 8} \\\ & \Rightarrow D= \sqrt{196-96} \\\ & \Rightarrow D= \sqrt{100} \\\ & \Rightarrow D=10 \\\ \end{aligned}$
So, after substituting the values of a, b, c we get the value 10 for discriminant.
Now, we know that solution is given by:
$sol=\dfrac{-b\pm D}{2a}$
Now, substituting the values in this formula we get,
$sol=\dfrac{-14\pm 10}{2\times 3}$
Now, we will get two values for x since it is a quadratic equation by taking +10 and -10.
$\begin{aligned} & sol=\dfrac{-14+10}{2\times 3} \\\ & \Rightarrow \dfrac{-4}{6} \\\ & \Rightarrow \dfrac{-2}{3} \\\ \end{aligned}$
This is one value and for the other value we will take discriminant as -10.
$\begin{aligned} & sol=\dfrac{-14-10}{2\times 3} \\\ & \Rightarrow \dfrac{-24}{6} \\\ & \Rightarrow -4 \\\ \end{aligned}$
And the other value is -4.
So, the values of x that are attained in the given question are -4 and $\dfrac{-2}{3}$ .
We can also make the factors of the given quadratic problem and then find the solution as said above and the factors would be $(x+4)$ and $(3x+2)$ .

Note: Trying to use the method wisely as sometimes it is difficult to make the factors of the quadratic equation and hence, we get into a mess as it becomes quite impossible to find values in this situation. Discriminant methods can be used to find the solutions in this case.