Matematika kelas X - Persamaan Kuadrat part 1 - Cara Pemfaktoran, Rumus ABC & Kuadrat Sempurna

Introduction

This tutorial focuses on solving quadratic equations, specifically through factoring, the use of the quadratic formula (Rumus ABC), and perfect squares. Understanding these concepts is essential for students in mathematics, particularly in grade X. By following this guide, you'll gain clarity on the methods to solve quadratic equations and enhance your problem-solving skills.

Step 1: Understanding Quadratic Equations

• A quadratic equation is in the standard form: [ ax^2 + bx + c = 0 ] where:
  • ( a ), ( b ), and ( c ) are constants
  • ( x ) represents the variable

Key Characteristics

• The degree of the equation is 2.
• The graph of a quadratic equation is a parabola.

Step 2: Factoring Quadratic Equations

Factoring is a method to solve quadratic equations by expressing them as a product of two binomials.

Steps to Factor

1. Identify values of ( a ), ( b ), and ( c ) from the quadratic equation.
2. Find two numbers that multiply to ( ac ) and add up to ( b ).
3. Rewrite the equation in factored form: [ a(x - p)(x - q) = 0 ] where ( p ) and ( q ) are the two numbers found.

Example

For the equation ( x^2 + 5x + 6 = 0 ):

• ( a = 1 ), ( b = 5 ), ( c = 6 )
• Two numbers that multiply to ( 6 ) (16 or 23) and add to ( 5 ) are ( 2 ) and ( 3 ).
• Factored form: [ (x + 2)(x + 3) = 0 ]

Step 3: Using the Quadratic Formula

The quadratic formula can solve any quadratic equation, especially when factoring is difficult.

Formula

[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ]

Steps to Apply the Formula

1. Identify ( a ), ( b ), and ( c ) from the equation.
2. Calculate the discriminant ( b^2 - 4ac ).
3. Substitute ( a ), ( b ), and the square root of the discriminant into the formula to find ( x ).

Example

For the equation ( 2x^2 + 4x - 6 = 0 ):

• ( a = 2 ), ( b = 4 ), ( c = -6 )
• Discriminant: [ 4^2 - 4(2)(-6) = 16 + 48 = 64 ]
• Calculate ( x ): [ x = \frac{-4 \pm \sqrt{64}}{2(2)} = \frac{-4 \pm 8}{4} ]
  • Solutions: ( x = 1 ) and ( x = -3 )

Step 4: Recognizing Perfect Squares

Perfect squares can simplify solving quadratics.

Perfect Square Form

• A quadratic expression is a perfect square if it can be expressed as: [ (x + a)^2 = x^2 + 2ax + a^2 ]

Steps to Identify

1. Check if the first and last terms are perfect squares.
2. Determine if the middle term is twice the product of the square roots of the first and last terms.

Example

For ( x^2 + 6x + 9 ):

• ( (x + 3)^2 = 0 )

Conclusion

In this tutorial, we explored different methods to solve quadratic equations, including factoring, using the quadratic formula, and recognizing perfect squares. Mastering these techniques will enhance your mathematical skills and prepare you for more complex problems. As a next step, practice solving various quadratic equations using these methods to solidify your understanding.