Grade 9 Mathematics: Quadratic equation.pptx

A PowerPoint presentation in grade 9 mathematics.

1. Quadratic Equations

2. At the end of the lesson, you should be able to: illustrate quadratic equations; convert from the standard form of a quadratic equation to its general form and vice versa; characterize the roots of a quadratic equation using the discriminant; and describe the relationship between the coefficients and the roots of a quadratic equation.

3. Quadratic Equation A quadratic equation is an equation that can be written in the form where , and are real numbers and . This form of a quadratic equation if called the standard form. The quadratic equation in one variable is an equation of degree 2, and the terms and are called quadratic term, linear term, and constant, respectively. Ex.   quadratic term   linear term   constant  

4. Quadratic Equation Ex. Rewrite the equation in standard form to determine if it is a quadratic equation. Then, identify the coefficient, linear term, and the constant.  

5. Roots of Quadratic Equation The value of x that will satisfy the equation is called the roots of the quadratic equation. It is also called the solution of the equation. A quadratic equation can have a maximum of two roots. The number and nature of roots of a quadratic equation can be identified by solving for the discriminant, . The discriminant is used to characterize the roots of quadratic equations without having to solve them.  

6. Roots of Quadratic Equation Given a quadratic equation , where : If , then the roots of the quadratic equation are equal and are both rational numbers, hence there is only one real and rational solution. If and is a perfect square, then there are two distinct real roots that are both rational numbers. If but is not a perfect square, then there are two distinct real roots that are both irrational If , then the quadratic equation has no real roots. If the discriminant is negative, the roots are said to be two distinct imaginary numbers.  

7. Roots of Quadratic Equation Determine the number and nature of the roots of the following quadratic equations by solving for their discriminants. - find the discriminant. - substitute a, b, and c. - simplify. Since , then the quadratic equation has no real root (has two distinct imaginary roots).  

8. Roots of Quadratic Equation Determine the number and nature of the roots of the following quadratic equations by solving for their discriminants. - standard form - find the discriminant. - substitute a, b, and c. - simplify. Since the discriminant is , then the quadratic equation has one real and rational solution.  

9. Sum and Product of the Roots of Quadratic Equation In general, let and be the roots of the quadratic equation , then the sum of its roots can be determined using the following formula: while the product of its roots can be computed by the formula:  

10. Sum and Product of the Roots of Quadratic Equation Example: Determine the sum and the product of the roots of the following quadratic equations. - write the equation in standard from. - identify the coefficients a and b and the constant c. Sum of roots: Product of roots: The sum of the roots of is 2, and the product of the roots is 1.  

11. Sum and Product of the Roots of Quadratic Equation Example: Determine the sum and the product of the roots of the following quadratic equations. - identify the coefficients a and b and the constant c. Sum of roots: Product of roots: The sum of the roots of is , and the product of the roots is .