Find the roots of each of the following quadratic equations by the method of completing the squares $\sqrt 5 x^2 + 9x + 4 \sqrt 5 = 0$
- $\displaystyle - \sqrt 7, \frac{-17}{\sqrt 3 }$
- $\displaystyle - \sqrt 5, \frac{-4}{\sqrt 5 }$
- $\displaystyle - \sqrt 5, \frac{-14}{\sqrt 3 }$
- $\displaystyle - \sqrt 7, \frac{-13}{\sqrt 5 }$
To solve sqrt(5)x^2 + 9x + 4sqrt(5) = 0, we can factor by grouping. We look for two numbers that multiply to (sqrt(5) * 4sqrt(5)) = 20 and add to 9. These are 5 and 4. The equation becomes sqrt(5)x^2 + 5x + 4x + 4sqrt(5) = 0. Factoring gives sqrt(5)x(x + sqrt(5)) + 4(x + sqrt(5)) = 0, which leads to (sqrt(5)x + 4)(x + sqrt(5)) = 0. The roots are -sqrt(5) and -4/sqrt(5).
To solve square root of 5 times x squared plus 9x plus 4 times square root of 5 equals 0 by completing the square, divide by the square root of 5 to get x squared plus 9 divided by the square root of 5 times x plus 4 equals 0. Moving the constant and adding the square of half the coefficient of x, which is (9 divided by 2 times square root of 5) squared or 81 divided by 20, yields (x plus 9 divided by 2 times square root of 5) squared equals 1 divided by 20. Taking the square root of both sides gives x plus 9 divided by 2 times square root of 5 equals 1 divided by 2 times square root of 5 or negative 1 divided by 2 times square root of 5. Solving these two equations results in the roots x equals negative 4 divided by square root of 5 and x equals negative square root of 5.