Questions: What does the fundamental theorem of algebra state about the equation (2 x^2+8 x+14=0) ?

Transcript text: What does the fundamental theorem of algebra state about the equation $2 x^{2}+8 x+14=0$ ?

Solution

Solution Steps

Step 1: Identify the equation

The given equation is $ 2x^{2} + 8x + 14 = 0 $. This is a quadratic equation of the form $ ax^{2} + bx + c = 0 $, where $ a = 2 $, $ b = 8 $, and $ c = 14 $.

Step 2: Apply the Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra states that every non-constant polynomial equation with complex coefficients has at least one complex root. Since the given equation is a quadratic polynomial, it must have exactly two roots (real or complex).

Step 3: Determine the nature of the roots

To determine whether the roots are real or complex, calculate the discriminant $ D $ using the formula: \[ D = b^{2} - 4ac \] Substitute the values of $ a $, $ b $, and $ c $: \[ D = 8^{2} - 4 \cdot 2 \cdot 14 = 64 - 112 = -48 \] Since the discriminant $ D $ is negative ($ D < 0 $), the equation has two complex conjugate roots.

Final Answer

The equation $ 2x^{2} + 8x + 14 = 0 $ has two complex conjugate roots. Thus, according to the Fundamental Theorem of Algebra, it has exactly two roots.

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