Algebra and Trigonometry
Algebraic Equations and Roots
Learning Objectives:
- Recognize the relationship between the roots of a quadratic equation and its coefficients
- Identify the sum and product of the roots.
- 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 \(\sqrt{b^2-4ac}\) part of this equation is called the discriminant. Based on its value, there are three possibilities:
If \(\sqrt{b^2-4ac}\) > 0 then the roots are real and distinct.
If \(\sqrt{b^2-4ac}\) = 0, then the roots are same (repeated) but real.
If \(\sqrt{b^2-4ac}\)< 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
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