Question

why is it necessary to use the quadratic formula when solving the equation $2x^2 + 4x + 5 = 0$ for complex solutions? \
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$\bigcirc$ because the discriminant is negative, indicating that the solutions are complex. \
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$\bigcirc$ because factoring is not possible when the coefficients are even numbers. \
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$\bigcirc$ because the quadratic formula is the only method that works for all types of quadratic equations. \
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$\bigcirc$ because completing the square is too complicated for this equation.

Explanation:

1. First, calculate the discriminant of the quadratic equation $ax^2+bx+c=0$, which is $\Delta = b^2-4ac$. For $2x^2+4x+5=0$, $a=2$, $b=4$, $c=5$.
2. A negative discriminant means the equation has no real solutions, only complex ones. Factoring over real numbers is not possible here, but the quadratic formula can handle complex solutions directly.
3. Evaluate the options:

• Factoring is not impossible just because coefficients are even; this is incorrect.
• The quadratic formula is not the only method (completing the square also works), so this is incorrect.
• Completing the square is not overly complicated for this equation, so this is incorrect.
• The discriminant being negative correctly explains why we need the quadratic formula for complex solutions here.

Answer:

A. Because the discriminant is negative, indicating that the solutions are complex.