Question

Solve the following quadratic equation 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: Compare the given quadratic equation with the general quadratic equation, and find the value of coefficients as variables a, b, c. Take the product of highest degree coefficient and constant. Use this value to factorize the equation. By using the factored form, try to get values of x satisfying the equation. These values of x are required to result in the question.

Complete step-by-step solution -
Factorization: For factorising a quadratic follow the steps:
Allot ${{x}^{2}}$ coefficient as “a”, x co-efficient as “b”, constant as “c”.
Find the product of the 2 numbers a, c. Let it be k.
Find 2 numbers such that their product is k, sum is b.
Rewrite the term $bx$ in terms of those 2 numbers.
Now, factor the first two terms.
Now, factor the last two terms.
If we do it correctly our 2 new terms will have one more common factor.
Now take that common to get quadratic as a product of 2 terms, from which you get the roots.
Given expression in the question can be written as:
$2{{x}^{2}}+7x+3=0$
Here, we get $a=2,b=7,c=3$ . so, product ac is:
$a\cdot c=2\cdot 3=6$
So, the 2 numbers whose product is 6, sum is 7 are:
1, 6 $\left( \because 1\times 6=6;1+6=7 \right)$
By this we can write our quadratic equation as:
$2{{x}^{2}}+6x+x+3=0$
Now take 2x as common from first two terms, we get:
$\Rightarrow 2x\left( x+3 \right)+x+3=0$
Now take $\left( x+3 \right)$ common from whole equation, we get:
$\Rightarrow \left( 2x+1 \right)\left( x+3 \right)=0$
By this we get 2 equations, which are written as:
$2x+1=0$ , $x+3=0$
From these we get the roots as given below:
$x=-\dfrac{1}{2},x=-3$
Therefore, option (b) is correct.

Note: The idea of getting the 2 numbers fitting the required condition is the only crucial element, do it properly. Instead of \[6x+x\] even if you write \[x+6x\], you take x common from first two terms and 3 as common from last two terms, even then you will reach the same result. So, it doesn’t matter in which order you use the 2 numbers.