Question

Solve the given quadratic equation $2{{x}^{2}}+7x+5=0$ .

Answer 1

Hint: Consider the given quadratic $2{{x}^{2}}+7x+5=0$ and write it as , $2{{x}^{2}}+2x+5x+5=0$ and then factorised as $\left( 2x+5 \right)\left( x+1 \right)=0$ and finally get values of x.

Complete step-by-step solution -
We are given a quadratic equation $2{{x}^{2}}+7x+5=0$ and we have to solve to find the value of x.
$2{{x}^{2}}+7x+5=0$ Is considered as quadratic equation is any equation that can be rearranged in standard form as $a{{x}^{2}}+bx+c=0$ .
Here x represents unknown and a, b, c are known numbers where $a\ne 0$. Otherwise, it becomes linear as no $a{{x}^{2}}$ term is there. The number a, b, c are coefficients of the equation and may be distinguished by calling them respectively, the quadratic coefficient, the linear coefficient, the linear coefficient, and the constant or free term.
The values of x that satisfy the equation are called the solution of the equation and roots or zeroes of the expression on its left-hand side. A quadratic equation has at most two solutions. If there is no real solution, one says it is double root. A quadratic equation always has two roots, If complex roots are included and a double root is counted for two. A quadratic equation of the form$a{{x}^{^{2}}}+bx+c=0$ can be factored as $a\left( x-\alpha \right)\left( x-\beta \right)=0$ where $\alpha $ and $\beta $ are solution of x.
Because the quadratic equation involves only one known, it is called coordinate. The quadratic equation only contains the power of x that are non – negative integers, and therefore, it is a polynomial equation. In particular, it is a second-degree polynomial equation.
As the equation is given as $2{{x}^{2}}+7x+5=0$
So, $2{{x}^{2}}+7x+5=0$ .
Can be written as,
$2{{x}^{2}}+2x+5x+5=0$ .
Which can be factorised as,
 $\left( x+1 \right)\left( 2x+5 \right)=0$ .
So , for the equation value -1 and $\dfrac{-5}{2}$ satisfies equation in place of x , Hence the value of x is -1 and $\dfrac{-5}{2}$ .

Note: We can also solve it using a formula or by Sridharacharya method which is $x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$ for quadratic equation $a{{x}^{2}}+bx+c=0$ . Here students should be careful while solving the question. Sign mistakes are very common and students should be cautious about them.