Question

Math

Posted 5 months ago

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Solve the equation $z^{\wedge} 2+4 z+13=0$, where $z$ is a complex number.
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Answer from Sia

Verified answer

Posted 5 months ago

Solution by Steps

step 1

To solve the quadratic equation $z^2 + 4z + 13 = 0$ for $z$, we use the quadratic formula $z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a = 1$, $b = 4$, and $c = 13$

step 2

Calculate the discriminant: $b^2 - 4ac = 4^2 - 4 \cdot 1 \cdot 13 = 16 - 52 = -36$

step 3

Since the discriminant is negative, the solutions will be complex numbers

step 4

Substitute the values into the quadratic formula: $z = \frac{-4 \pm \sqrt{-36}}{2 \cdot 1} = \frac{-4 \pm 6i}{2}$

step 5

Simplify the expression: $z = \frac{-4}{2} \pm \frac{6i}{2} = -2 \pm 3i$

Answer

The solutions to the equation $z^2 + 4z + 13 = 0$ are $z = -2 + 3i$ and $z = -2 - 3i$.

Key Concept

Solving quadratic equations with complex solutions

Explanation

When the discriminant of a quadratic equation is negative, the solutions are complex numbers. The quadratic formula can be used to find these solutions.

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