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Find the roots of the following quadratic (if they exist) by the method of completing the square.

\(\sqrt{3}x^2+10x+7\sqrt{3}=0\)

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Given: \(\sqrt{3}x^2+10x+7\sqrt{3}=0\)

To find: the roots of the following quadratic (if they exist) by the method of completing the square.

Solution: We have to make the quadratic equation a perfect square if possible or sum of perfect square with a constant.

Step 1: Make the coefficient of x2 unity.In the equation

\(\sqrt{3}x^2+10x+7\sqrt{3}=0\),

The coefficient of x2 is \(\sqrt{3}\)

So to make the coffecient of x2 equals to 1.

divide the whole equation by \(\sqrt{3}\) .

The quadratic equation now becomes:

Step 2: Shift the constant term on RHS,

Step 3: Add square of half of coefficient of x on both the sides.

Step 4: Apply the formula, (a + b)2 = a2 + 2ab + b2 on LHS and solve RHS,

Here a = x and \(b = \frac{5}{\sqrt{3}}\)

As RHS is positive, the roots exist.

Step 5: take square root on both sides,

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