Question

How do you solve $2{{x}^{2}}+3x-14=0$ ?

Answer 1

Hint: In this question, we have to find the value of x. The equation given to us is in the form of a quadratic, therefore, we will apply the discriminant method to solve this problem. We will first compare the given equation with the general form of the quadratic equation and thus get the value of a, b, and c. Then, we will find the value of discriminant $D=\sqrt{{{b}^{2}}-4ac}$, and then apply the discriminant formula $x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$ . After the necessary calculations, we get two equations of x, so we solve them separately to get the value of x, which is our required answer.

Complete step by step solution:
According to the question, a quadratic equation is given to us and we have to solve the equation for the value of x.
The equation is $2{{x}^{2}}+3x-14=0$ ----------------- (1)
As we know, the general quadratic equation is in form of $a{{x}^{2}}+bx+c=0$ ---------- (2)
Thus, on comparing equation (1) and (2), we get $a=2,$ $b=3,$ and $c=-14$ ------- (3)
So, now we will apply the discriminant formula $D=\sqrt{{{b}^{2}}-4ac}$ by putting the above values in the formula, we get
$\begin{align}
  & \Rightarrow D=\sqrt{{{(3)}^{2}}-4.(2).(-14)} \\
 & \Rightarrow D=\sqrt{9+112} \\
\end{align}$
Thus, on further solving, we get
$\Rightarrow D=\sqrt{121}$
We see that the square root has a positive term which implies the discriminant has real roots, thus we get
$\Rightarrow D=11$ -------------- (4)
Now, we will apply the discriminant formula , which is
$\Rightarrow x=\dfrac{-b\pm D}{2a}$
$\Rightarrow x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$ --------------- (5)
So, we will put the value of equation (3) and (4) in equation (5), we get
$\Rightarrow x=\dfrac{-(3)\pm 11}{2.(2)}$
On further simplification, we get
$\Rightarrow x=\dfrac{-3\pm 11}{4}$
Therefore, we will split the above equation in terms of (+) and (-) sign, we get
$\Rightarrow x=\dfrac{-3+11}{4}$ -------- (6) , or
 $\Rightarrow x=\dfrac{-3-11}{4}$ ---------- (7)
Now, we will first solve equation (6) that is we will split the denominator with respect to addition, we get
$\Rightarrow x=\dfrac{-3}{4}+\dfrac{11}{4}$
On taking the LCM in the above equation, we get
$\Rightarrow x=\dfrac{-3+11}{4}$
On further simplification, we get
$\begin{align}
  & \Rightarrow x=\dfrac{8}{4} \\
 & \Rightarrow x=2 \\
\end{align}$
Now, we will first solve equation (7) that is we will split the denominator with respect to subtraction, we get
$\Rightarrow x=\dfrac{-3}{4}-\dfrac{11}{4}$
On taking the LCM in the above equation, we get
$\Rightarrow x=\dfrac{-3-11}{4}$
On further simplification, we get
$\begin{align}
  & \Rightarrow x=\dfrac{-14}{4} \\
 & \Rightarrow x=-\dfrac{7}{2} \\
\end{align}$

Therefore, for the equation $2{{x}^{2}}+3x+14=0$ , we get the value of $x=2,-\dfrac{7}{2}$

Note: While solving this problem, do all the steps carefully and avoid errors to get the correct answer. You can also use splitting the middle term method or the cross multiplication method to get the required solution to the problem.