Persamaan Kuadrat Dan Akar-akar Persamaan Kuadrat | Matematika Kelas 9

Introduction

This tutorial aims to provide a comprehensive guide on quadratic equations and their roots, tailored for 9th-grade mathematics students. Understanding quadratic equations is essential for solving various mathematical problems and lays the groundwork for advanced topics in algebra. This guide will walk you through key concepts and examples related to quadratic equations and their roots.

Step 1: Understanding Quadratic Equations

• A quadratic equation is a polynomial equation of the form

  [ ax^2 + bx + c = 0 ] where:
  • ( a ), ( b ), and ( c ) are constants (with ( a \neq 0 )).
  • ( x ) represents an unknown variable.

• Quadratic equations can be identified by their highest exponent of 2.

Step 2: Identifying Roots of Quadratic Equations

• The roots (or solutions) of a quadratic equation are the values of ( x ) that satisfy the equation.

To find the roots, you can use several methods, including

• Factoring
• Completing the square

Quadratic formula

[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ]
• The expression under the square root (( b^2 - 4ac )) is known as the discriminant.

Step 3: Exploring the Discriminant

The discriminant helps determine the nature and number of roots

• If ( b^2 - 4ac > 0 ): two distinct real roots.
• If ( b^2 - 4ac = 0 ): one real root (repeated).
• If ( b^2 - 4ac < 0 ): no real roots (two complex roots).

Step 4: Example Problems

Example 1: Identifying a Quadratic Equation

Given the equation ( 3x^2 + 2x - 5 = 0 ), determine if it's a quadratic equation

• It is a quadratic equation as it follows the standard form.

Example 2: Finding Roots

For the equation ( 2x^2 - 4x + 2 = 0 )

Calculate the discriminant

[ b^2 - 4ac = (-4)^2 - 4(2)(2) = 16 - 16 = 0 ]
• Since the discriminant equals zero, there is one repeated root.

Example 3: Determining Number of Roots

For the equation ( x^2 + 4x + 8 = 0 )

Calculate the discriminant

[ b^2 - 4ac = (4)^2 - 4(1)(8) = 16 - 32 = -16 ]
• Since the discriminant is negative, there are no real roots.

Example 4: Solving for Variables

Find the value of ( m ) in ( mx^2 - 6x + 9 = 0 ) that ensures one real root

Set the discriminant to zero

[ (-6)^2 - 4(m)(9) = 0 \implies 36 - 36m = 0 \implies m = 1 ]

Conclusion

In this tutorial, you learned about quadratic equations, their roots, and how to analyze them using the discriminant. Practice identifying and solving quadratic equations using the methods discussed. For further study, explore additional resources or exercises on quadratic equations to enhance your skills.