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If using the method of completing the square to solve the quadratic equation x2 + x +9 = 0, which number would have to be added to "complete the square"?

User Antibus
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1 Answer

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The given equation is


x^2+x+9=0

To complete the square, first, we have to move the independent term to the other side.


x^2+x=-9

Then, we divide the linear coefficient by 2, then we apply a square power to it.


((1)/(2))^2=(1)/(4)

We add this fraction to each side.


x^2+x+(1)/(4)=-9+(1)/(4)

Then, we factor the trinomial and sum the numbers on the right side.


\begin{gathered} (x+(1)/(2))^2=(-36+1)/(4) \\ (x+(1)/(2))^2=(-35)/(4) \end{gathered}

Then, to solve for x, we use a square root on both sides.


\begin{gathered} \sqrt[]{(x+(1)/(2))^2}=\pm\sqrt[]{-(35)/(4)} \\ (x+(1)/(2))=\pm\sqrt[]{-(35)/(4)} \end{gathered}

As you can observe, the equation has no real solutions because the square root of a negative number can be solved in the real numbers.

Hence, the equation has no real solutions, and the number added to complete the square is 1/4.

User Nelani
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