Question

Solve the given equation by complete square method $2{{x}^{2}}+3x-5=0$

Answer 1

Hint: To solve the given quadratic equation by complete square method first we will divide the whole equation by a. Now we have an equation of the form ${{x}^{2}}+\dfrac{b}{a}+\dfrac{c}{a}=0$ . Next we add and subtract a term such that we get a perfect square of the form ${{\left( a+b \right)}^{2}}$ on LHS . Now taking the constant on RHS and then taking square root we will get the required value of x.

Complete step by step answer:
Now consider the given equation $2{{x}^{2}}+3x-5=0$
Now the given equation is a quadratic equation of the form $a{{x}^{2}}+bx+c=0$ where a = 2, b = 3 and c = - 5.
Now let us divide the whole equation by a which is 2. Hence, we get
${{x}^{2}}+\dfrac{3}{2}x-\dfrac{5}{2}=0$
Now adding and subtracting $\dfrac{9}{16}$ on LHS we get
$\implies {{x}^{2}}+\dfrac{3}{2}x+\dfrac{9}{16}-\dfrac{9}{16}-\dfrac{5}{2}=0$
$\implies {{x}^{2}}+\dfrac{3}{2}x+\dfrac{9}{16}-\dfrac{49}{16}=0$
Now writing the above equation in the form ${{a}^{2}}+2ab+{{b}^{2}}$ and taking the constant to RHS we get,
${{x}^{2}}+2\times \dfrac{3}{4}\times 1+{{\left( \dfrac{3}{4} \right)}^{2}}=\dfrac{49}{16}$
Now we know that the expansion of ${{\left( a+b \right)}^{2}}$ is ${{a}^{2}}+2ab+{{b}^{2}}$
Hence we can write \[{{x}^{2}}+2\left( \dfrac{3}{4} \right)\left( 1 \right)+{{\left( \dfrac{3}{4} \right)}^{2}}\] as ${{\left( x+\dfrac{3}{4} \right)}^{2}}$
Therefore we have ${{\left( x+\dfrac{3}{4} \right)}^{2}}=\dfrac{49}{16}$
Taking square root on both sides we get,
$\left( x+\dfrac{3}{4} \right)=\pm \dfrac{7}{4}$
Now we have two solutions,
Either $x+\dfrac{3}{4}=\dfrac{7}{4}$ or $x+\dfrac{3}{4}=-\dfrac{7}{4}$
Taking the terms on RHS we get,
$x=\dfrac{7}{4}-\dfrac{3}{4}$ or $x=-\dfrac{7}{4}-\dfrac{3}{4}$
Hence the values of x are x = 1 or $x=-\dfrac{10}{4}=-\dfrac{5}{2}$
Hence the solution of the given quadratic equation is x = 1 or $x=-\dfrac{5}{2}$ .

Note: Now note that when finding the term which is supposed to be added and subtracted to the equation to form a complete square divide the middle term which is $\dfrac{b}{a}$ by 2 and square. Hence the term which needs to be added and subtracted is ${{\left( \dfrac{b}{2a} \right)}^{2}}$.