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# Find the roots of the following quadratic equation (if they exist) by method of completing square method.$2{{x}^{2}}+x+4=0$

Last updated date: 26th Mar 2023
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Hint: To find the roots first you must check whether the roots are real or imaginary. For that use $D={{b}^{2}}-4ac$. Check it, you will get the answer.

In Mathematics, an equation is a statement that asserts the equality of two expressions. The word equation and its cognates in other languages may have subtly different meanings; for example, in French an equation is defined as containing one or more variables, while in English any equality is an equation.

Solving an equation containing variables consists of determining which values of the variables make the equality true. Variables are also called unknowns and the values of the unknowns that satisfy the equality are called solutions of the equation. There are two kinds of equations: identities and conditional equations. An identity is true for all values of the variable. A conditional equation is only true for particular values of the variables.

An equation is written as two expressions, connected by an equals sign ("$=$"). The expressions on the two sides of the equals sign are called the "left-hand side" and "right-hand side" of the equation.

The most common type of equation is an algebraic equation, in which the two sides are algebraic expressions. Each side of an algebraic equation will contain one or more terms. For example, the equation.
$A{{x}^{2}}+Bx+C=y$

has left-hand side $A{{x}^{2}}+Bx+C$ which has three terms, and right-hand side $y$, consisting of just one term. The unknowns are $x$ and $y$ and the parameters are $A,$ $B,$ and $C$.

An equation is analogous to a scale into which weights are placed. When equal weights of something (grain for example) are placed into the two pans, the two weights cause the scale to be in balance and are said to be equal. If a quantity of grain is removed from one pan of the balance, an equal amount of grain must be removed from the other pan to keep the scale in balance. Likewise, to keep an equation in balance, the same operations of addition, subtraction, multiplication and division must be performed on both sides of an equation for it to remain true.

Now, we are given an equation $2{{x}^{2}}+x+4=0$.

So, let us check whether the roots of the equation exist or not.

Here, $a=2,$ $b=1$ and $c=4$.

$D={{b}^{2}}-4ac$

Substituting the values,

$D={{(1)}^{2}}-4(2)(4)$

Simplifying we get,

$D=1-32=-31$
So here we can see that roots are unequal and imaginary.

We cannot solve it by completing the square method.

Note: Read the question carefully. Be thorough with the concepts. You should know the nature of roots i.e. when the roots are real, imaginary etc. Also, while finding roots do not miss any term. Also, don’t miss any sign. Take care that no mistakes occur.