Learning Objectives:

1. Recognize the relationship between the roots of a quadratic equation and its coefficients
2. Identify the sum and product of the roots.
3. Form a quadratic equation given its roots, including when the roots are integers, rational numbers, real numbers, and complex numbers, and solve related problems.

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Quadratic Equation:

Any equation in the form of \(ax^2 + bx + c = 0\) where a\(\neq\) 0 is referred as quadratic eqation and its roots are given by the formula: \[x_{1,2} = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

The \(\sqrt{b^2-4ac}\) part of this equation is called the discriminant. Based on its value, there are three possibilities:

1. If \(\sqrt{b^2-4ac}\) > 0 then the roots are real and distinct.

2. If \(\sqrt{b^2-4ac}\) = 0, then the roots are same (repeated) but real.

3. If \(\sqrt{b^2-4ac}\)< 0, then the roots have imaginary parts and they are complex numbers.

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Solved Example: 1-11-01

Roots of a quadratic equation are imaginary when discriminant is ______:

A. Zero

B. Greater than zero

C. Less than zero

D. Greater than or equal to zero

Correct Answer: C

Solved Example: 1-11-02

If $x^2−14x+k = 0$ is a perfect square, then k=?

A. -7

B. 196

C. -49

D. 49

Correct Answer: D