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What is a complex number?

Back

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit, defined as the square root of -1.

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What is the discriminant in a quadratic equation?

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The discriminant is the part of the quadratic formula under the square root, given by b² - 4ac, which determines the nature of the roots of the quadratic equation ax² + bx + c = 0.

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What does a positive discriminant indicate about the solutions of a quadratic equation?

Back

A positive discriminant indicates that the quadratic equation has two distinct real solutions.

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What does a zero discriminant indicate about the solutions of a quadratic equation?

Back

A zero discriminant indicates that the quadratic equation has exactly one real solution (a repeated root).

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What does a negative discriminant indicate about the solutions of a quadratic equation?

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A negative discriminant indicates that the quadratic equation has two complex (non-real) solutions.

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What is the quadratic formula?

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The quadratic formula is x = (-b ± √(b² - 4ac)) / (2a), used to find the solutions of a quadratic equation.

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How do you simplify the expression (8 + 5i) / (1 - 7i)?

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To simplify (8 + 5i) / (1 - 7i), multiply the numerator and denominator by the conjugate of the denominator: (1 + 7i). The result is (-27 + 61i) / 50.

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What is the conjugate of a complex number a + bi?

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The conjugate of a complex number a + bi is a - bi.

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How do you add two complex numbers?

Back

To add two complex numbers, add their real parts and their imaginary parts separately: (a + bi) + (c + di) = (a + c) + (b + d)i.

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How do you subtract two complex numbers?

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To subtract two complex numbers, subtract their real parts and their imaginary parts separately: (a + bi) - (c + di) = (a - c) + (b - d)i.