Kelas 10 Bab 5 | Persamaan dan Fungsi kuadrat

Introduction

In this tutorial, we will explore the concepts of quadratic equations and functions, specifically tailored for 10th-grade mathematics. Through this guide, you will learn the definitions, methods for solving quadratic equations, and key properties like the discriminant and maximum values. This knowledge will help you tackle school assignments and prepare for examinations.

Step 1: Understanding Quadratic Equations

• Definition: A quadratic equation is a polynomial equation of the form ( ax^2 + bx + c = 0 ), where ( a ), ( b ), and ( c ) are constants, and ( a \neq 0 ).
• Key Characteristics:
  • The highest power of the variable ( x ) is 2.
  • The graph of a quadratic function is a parabola.

Step 2: Solving Quadratic Equations by Factorization

• Factorization Method:

  1. Write the equation in standard form: ( ax^2 + bx + c = 0 ).
  2. Find two numbers that multiply to ( ac ) and add up to ( b ).
  3. Rewrite the middle term using these two numbers.
  4. Factor by grouping.
  5. Set each factor to zero and solve for ( x ).

• Example: For ( x^2 + 5x + 6 = 0 ):

  • Factor as ( (x + 2)(x + 3) = 0 )
  • Solutions: ( x = -2 ) and ( x = -3 ).

Step 3: Solving Perfect Square Quadratic Equations

• Perfect Square Form: A quadratic equation can be expressed as ( (px + q)^2 = 0 ).

• Steps:

  1. Identify if the quadratic can be rewritten as a perfect square.
  2. Solve the equation ( (px + q)^2 = 0 ).
  3. Find the value of ( x ).

• Example: For ( x^2 + 4x + 4 = 0 ):

  • Rewrite as ( (x + 2)^2 = 0 )
  • Solution: ( x = -2 ).

Step 4: Using the Quadratic Formula

• Quadratic Formula: The solutions to ( ax^2 + bx + c = 0 ) can be found using: [ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} ]

• Steps:

  1. Identify coefficients ( a ), ( b ), and ( c ).
  2. Calculate the discriminant ( D = b^2 - 4ac ).
  3. Use the quadratic formula to find ( x ).

• Example: For ( 2x^2 + 4x - 6 = 0 ):

  • Here, ( a = 2 ), ( b = 4 ), and ( c = -6 ).
  • Calculate ( D = 4^2 - 4 \cdot 2 \cdot (-6) = 16 + 48 = 64 ).
  • Solutions: ( x = \frac{{-4 \pm 8}}{{4}} ), yielding ( x = 1 ) and ( x = -3 ).

Step 5: Understanding the Discriminant

• Definition: The discriminant ( D = b^2 - 4ac ) determines the nature of the roots of the quadratic equation.
• Cases:
  • If ( D > 0 ): Two distinct real roots.
  • If ( D = 0 ): One real root (repeated).
  • If ( D < 0 ): No real roots (complex roots).

Step 6: Finding Maximum and Minimum Values of Quadratic Functions

• Vertex Formula: The maximum or minimum value of a quadratic function ( f(x) = ax^2 + bx + c ) occurs at ( x = -\frac{b}{2a} ).

• Steps:

  1. Calculate the vertex ( x ).
  2. Substitute ( x ) back into the function to find the corresponding ( y ) value.

• Example: For ( f(x) = -x^2 + 4x + 1 ):

  • Vertex ( x = -\frac{4}{2(-1)} = 2 ).
  • Maximum value ( f(2) = -2^2 + 4(2) + 1 = 5 ).

Step 7: Solving New Quadratic Equations with Known Roots

• Formulation: If the roots ( r_1 ) and ( r_2 ) are known, the quadratic equation can be expressed as: [ f(x) = a(x - r_1)(x - r_2) ]

• Steps:

  1. Expand the equation.
  2. Set it to standard form ( ax^2 + bx + c = 0 ).

• Example: Given roots ( 3 ) and ( -2 ):

  • Equation: ( f(x) = a(x - 3)(x + 2) ).
  • Expand to find ( f(x) = ax^2 - ax + 6a ).

Conclusion

In this tutorial, we covered the essential concepts of quadratic equations and functions, including definitions, solving methods, discriminants, and maximum values. Understanding these principles will enhance your mathematical skills and help you excel in exams. For further practice, solve various quadratic problems using the methods discussed. Happy studying!