Questions: Use the quadratic formula to solve the equation. 2 x^2 + 2 x + 7 = 0 x = □

Transcript text: Use the quadratic formula to solve the equation. \[ \begin{array}{l} 2 x^{2}+2 x+7=0 \\ x=\square \end{array} \]

Solution

Solution Steps

Step 1: Identify the coefficients

The coefficients are \(a = 2\), \(b = 2\), and \(c = 7\).

Step 2: Calculate the discriminant

The discriminant, \(b^2 - 4ac\), is calculated as \(b^2 - 4 \cdot 2 \cdot 7 = -52\).

Step 3: Determine the nature of the roots

Since the discriminant is less than 0, the equation has two complex roots.

Step 4: Calculate the roots using the quadratic formula

The roots are calculated using the formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Substituting the values, we get \(x_1 = (-0.5+1.8j)\) and \(x_2 = (-0.5-1.8j)\).

Final Answer:

The roots of the equation \(ax^2 + bx + c = 0\) are \(x_1 = (-0.5+1.8j)\) and \(x_2 = (-0.5-1.8j)\).

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