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Quadratic equations are a type of polynomial equation of the form \(ax^2 + bx + c = 0\). The highest exponent of the variable (usually x) is 2 in a quadratic equation. Here, 'a', 'b', and 'c' are constants where 'a' is non-zero.
Quadratic equations can have different types of solutions:

• Two distinct real roots
• One real root
• Two complex roots

Solving quadratic equations can involve factoring, completing the square, or using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). When the equation has complex roots, they typically appear as conjugates.

Complex numbers expand our understanding of numbers by adding a dimension beyond the real number line. They are of the form \(a + bi\) where 'a' and 'b' are real numbers and 'i' is the imaginary unit defined by \(i^2 = -1\). This allows us to solve equations that don't have real solutions.
For example, the equation \(x^2 + 1 = 0\) has no real solutions. However, if we use complex numbers, the solutions are \(i\) and \(-i\). Complex numbers are often used in engineering, physics, and applied mathematics.
To add or subtract complex numbers, combine their real parts and imaginary parts separately:

• Adding: \((a + bi) + (c + di) = (a+c) + (b+d)i\)
• Subtracting: \((a + bi) - (c + di) = (a-c) + (b-d)i\)

Multiplying complex numbers involves using the distributive property and the fact that\(i^2 = -1\):
\((a + bi)(c + di) = ac + adi + bci + bdi^2 = (ac - bd) + (ad + bc)i\)

In quadratic equations, when dealing with complex roots, they occur as conjugate pairs. The complex conjugate of \(a + bi\) is \(a - bi\). This property ensures that the coefficients of the equation remain real.
For example, if \(4i\) is a root of a quadratic equation, \(-4i\) must also be a root. Complex conjugates have the same real part but opposite imaginary parts.
When forming quadratic equations from given roots, such as \(4i\) and \(-4i\), we use the factored form \((x - r1)(x - r2) = 0\). This simplifies to:

• \((x - 4i)(x + 4i) = 0\)
• \(x^2 - (4i)^2 = x^2 - (16)(-1) = x^2 + 16\)

Thus, the quadratic equation with roots \(4i\) and \(-4i\) is \(x^2 + 16 = 0\). Understanding conjugate roots is crucial for solving quadratic equations with complex solutions.