Solving Quadratics with Imaginary Numbers

Front

What is the value of \( \sqrt{-1} \)?

Back

The value of \( \sqrt{-1} \) is \( i \), which is the imaginary unit.

Front

What is the result of squaring the imaginary number \( i\sqrt{5} \)?

Back

The result of squaring \( i\sqrt{5} \) is \( -5 \).

Front

What is the solution to the equation \( 36x^2 + 169 = 0 \)?

Back

The solution to the equation is \( x = \pm \frac{13}{6}i \).

Front

What does \( i^2 \) equal?

Back

\( i^2 = -1 \).

Front

What is the definition of an imaginary number?

Back

An imaginary number is a number that can be written as a real number multiplied by the imaginary unit \( i \), where \( i = \sqrt{-1} \).

Front

What is the general form of a quadratic equation?

Back

The general form of a quadratic equation is \( ax^2 + bx + c = 0 \), where \( a, b, \) and \( c \) are constants.

Front

How do you find the roots of a quadratic equation using the quadratic formula?

Back

The roots of a quadratic equation can be found using the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).

Front

What is the discriminant in a quadratic equation, and what does it indicate?

Back

The discriminant is given by \( b^2 - 4ac \). It indicates the nature of the roots: if positive, there are two distinct real roots; if zero, there is one real root; if negative, there are two complex roots.

Front

What is the significance of complex numbers in solving quadratic equations?

Back

Complex numbers allow for the solutions of quadratic equations that do not have real roots, particularly when the discriminant is negative.

Front

What is the relationship between imaginary numbers and complex numbers?

Back

Imaginary numbers are a subset of complex numbers, which are expressed in the form \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part.