Persamaan Kuadrat (5) - Membuat Persamaan Kuadrat Baru, Menyusun Persamaan Kuadrat - Matematika SMP

Introduction

This tutorial will guide you through creating quadratic equations, as discussed in the video "Persamaan Kuadrat (5) - Membuat Persamaan Kuadrat Baru." Understanding how to formulate quadratic equations is essential for students in junior high school mathematics, especially under the Merdeka Curriculum. We will cover various methods to construct quadratic equations based on given roots or relationships.

Step 1: Understanding Quadratic Equations

• A quadratic equation is generally expressed as: [ ax^2 + bx + c = 0 ]
• Here, (a), (b), and (c) are constants, and (x) represents the variable.
• The roots of the equation can be found using the quadratic formula: [ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ]

Step 2: Constructing a Quadratic Equation with Known Roots

• If you know the roots (let's call them (r_1) and (r_2)), you can use the factored form of the quadratic equation: [ (x - r_1)(x - r_2) = 0 ]
• To expand this, use the distributive property:
  • Multiply (x) by both terms:
    • (x^2 - (r_1 + r_2)x + r_1r_2 = 0)
• Therefore, the quadratic equation becomes: [ x^2 - (r_1 + r_2)x + r_1r_2 = 0 ]
• Practical Tip: Always check your roots by substituting them back into the equation.

Step 3: Creating a Quadratic with Roots as Twice Another Equation's Roots

• If you have another quadratic equation with roots (a_1) and (a_2), and want the new roots to be (2a_1) and (2a_2):
  • Start from the original equation: [ (x - a_1)(x - a_2) = 0 ]
  • Substitute (x) with (x/2) to scale the roots: [ \left(\frac{x}{2} - a_1\right)\left(\frac{x}{2} - a_2\right) = 0 ]
  • Expand and rearrange to get: [ x^2 - (2a_1 + 2a_2)x + 4a_1a_2 = 0 ]

Step 4: Constructing a Quadratic with Roots of Opposite Signs

• For roots (r_1) and (-r_2):
  • Use the factored form: [ (x - r_1)(x + r_2) = 0 ]
  • This expands to: [ x^2 + (r_2 - r_1)x - r_1r_2 = 0 ]
• Practical Tip: Ensure you keep track of the signs when expanding.

Step 5: Formulating Quadratics with Reciprocal Roots

• If you want to create a quadratic equation with roots that are reciprocals of another equation's roots (a_1) and (a_2):
  • Start with the original roots: [ (x - a_1)(x - a_2) = 0 ]
  • The new equation becomes: [ (ax - 1)(bx - 1) = 0 ]
  • Expanding gives: [ abx^2 - (a + b)x + 1 = 0 ]

Conclusion

In this tutorial, you learned how to create quadratic equations based on various conditions such as known roots, relationships between roots, and transformations. Practice these methods with different sets of roots to strengthen your understanding. For further study, explore more advanced topics in quadratic equations, such as their graphing and applications in real-world scenarios.