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You may recall that the square root of a negative number is non-real, because any real number squared will not be negative. The square root of -1 is defined as the imaginary unit i .
Imaginary numbers often arise when solving quadratic equations.
A complex number is a value in the following form, in which the variables a and b are real numbers and i is the imaginary unit.
Notice that in this standard form for writing a complex number, the real part a is written first and the imaginary part bi is written second.
EXAMPLE
In the complex number below, 5 is the real part of the complex number, and 3i is the imaginary part. Furthermore, in the imaginary part, 3 is the coefficient and i is the imaginary unit.Complex numbers can occur when solving a quadratic equation using the quadratic formula.
When solving a quadratic equation set equal to zero, as shown below, the solution(s), x, to the quadratic equation can be found using the quadratic formula.
The variables a, b, and c in the quadratic formula correspond to the coefficients in the quadratic equation.
If the expression under the square root is negative, then the quadratic equation will have zero real solutions. It follows, then, that when there are no real solutions to a quadratic equation, the graph of the equation will have zero x-intercepts, meaning that the parabola will never intersect the x-axis. In cases such as this, you can use the imaginary unit i to write the solutions of the quadratic equation as complex numbers.
EXAMPLE
Suppose you want to solve the quadratic equation:
Source: This work is adapted from Sophia author Colleen Atakpu.