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.

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

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

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

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

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

\[ax^2 + bx + c = 0\] Then $b^2-4ac$ is called the discriminant.

• If the discriminat is positive, the roots are real and distinct.
• If the discriminat is zero, the roots are real but equal.
• If the discriminat is negative, the roots are imaginary and complex conjugates of each other.

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

Focus on the middle term coefficient.
If you take half of middle term coefficient, $\dfrac{14}{2} = 7$, then we can make the original expression $(x - 7)^2$ as still have the middle term as -14x.
In such case, the last term can be found of by expanding, \[(x -7)^2 = x^2 - 14x + 49\]

Correct Answer: D

Solved Example: 1-11-03

If \[3x^2 - ax + 6 = ax^2 + 2x + 2\] has only one (repeated) solution, then the positive integral solution of a is:

A. 3

B. 2

C. 4

D. 5

Correct Answer: B

Solved Example: 1-11-04

One of the roots of the equation $x^2 - 12x + k = 0$ is x = 3. The other root is:

A. x = -4

B. x = 9

C. x = 4

D. x = -9

First substitute x = 3 in the given quadratic equation.
k = 27, Then use the quadratic formula.

Correct Answer: B