Question

Complete the square to solve the equation below.x2+x=1O A. x= 3; x = -1O B. x= -2+ V5; x = -2- 15C. x= - 1+ 15;x= - 3 - 5D. x= 1+ 15; x = 1 - 5

Answer 1

Consider the given quadratic equation,

`x^2+x=(19)/(4)`

Here the coefficient of 'x' is positive, so the algebraic identity which will be used in the method of completing squares, is as follows,

`a^2+2ab+b^2=\mleft(a+b\mright)^2`

Writing the left side of the given equation in the form of the left side expression of the identity,

`(x)^2+2(x)((1)/(2))=(19)/(4)`

It is observed that,

`\begin{gathered} a=x \\ b=(1)/(2) \end{gathered}`

Note that, in order to complete the square, the left side expression requires a 'b-squared' term.

This can be done by adding the term to both sides of the equality.

Adding the term both sides,

`\begin{gathered} (x)^2+2(x)((1)/(2))+((1)/(2))^2=(19)/(4)+((1)/(2))^2 \\ (x)^2+2(x)((1)/(2))+((1)/(2))^2=(19)/(4)+((1)/(4))^{} \\ (x)^2+2(x)((1)/(2))+((1)/(2))^2=(19+1)/(4) \\ (x)^2+2(x)((1)/(2))+((1)/(2))^2=5 \end{gathered}`

Now, the left side of the expression perfectly fits the left side expression of the algebraic identity, which can be replaced by its right side term,

`(x+(1)/(2))^2=5`

Taking square rootsboth sides,

`\begin{gathered} x+(1)/(2)=\pm\sqrt[]{5} \\ x+(1)/(2)=\sqrt[]{5}\text{ }or\text{ }x+(1)/(2)=-\sqrt[]{5}\text{ } \\ x=\sqrt[]{5}-(1)/(2)or\text{ }x=-\sqrt[]{5}-(1)/(2) \end{gathered}`

Thus, the solutions of the given quadratic equations are,

`x=\sqrt[]{5}-(1)/(2)or\text{ }x=-\sqrt[]{5}-(1)/(2)`

Therefore, option C is the correct choice.