Question

Solve and find the value of x: \[2{x^2} + 7x + 3 = 0\]
A.\[x = \dfrac{{ - 1}}{2},3\]
B.\[x = \dfrac{{ - 1}}{2}, - 3\]
C.\[x = \dfrac{1}{2}, - 3\]
D.\[x = \dfrac{1}{2},3\]

Answer 1

Hint: Here we have to use the concept of quadratic equations. We will first factorize the given equation. Then we will factor out the common terms. Then by applying zero product property we will get the value of \[x\].

Complete step by step solution:
We will first factorize the given equation for that we will split the term \[7x\] into two terms such that their product is equal to the product of first term and the last term. Therefore, we get
\[2{x^2} + 6x + x + 3 = 0\]
Now we will factor out \[2x\] from the first two terms. Therefore, we get
\[ \Rightarrow 2x\left( {x + 3} \right) + 1\left( {x + 3} \right) = 0\]
Now, we will take \[\left( {x + 3} \right)\] common from the equation. Therefore, we get
\[ \Rightarrow \left( {2x + 1} \right)\left( {x + 3} \right) = 0\]
Applying zero product property on the above equation, we get
\[\begin{array}{l} \Rightarrow 2x + 1 = 0\\ \Rightarrow 2x = - 1\end{array}\]
Dividing both the side by 2, we get
\[ \Rightarrow x = \dfrac{{ - 1}}{2}\]
Or
\[ \Rightarrow x + 3 = 0\]
Subtracting 3 from bith the side, we get
\[ \Rightarrow x = - 3\]
Hence, \[x\] is equal to \[\dfrac{{ - 1}}{2}\] and \[ - 3\].
So, option B is the correct option.

Note: The given equation is a quadratic equation. An equation is said to be a quadratic equation if the highest degree of the variable is 2. The degree 2 suggests that there are 2 roots of the given quadratic equation. Roots of the equations are those values, which when substituted in the equation, then the equation becomes zero. Therefore we have to form the equations in such a way that we will get the value of the variable \[x\]. The value of \[x\] are the roots of the equation.
We have also applied zero product property. Zero product property states that if \[a \cdot b = 0\], then either \[a = 0\] or \[b = 0\].