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Introduction

This tutorial will guide you through the process of finding the roots of quadratic equations using the method of factoring. Understanding this method is essential for solving quadratic equations efficiently, especially in algebra. Factoring is a valuable skill that can simplify complex problems and enhance your mathematical toolkit.

Step 1: Identify the Quadratic Equation

• Ensure the equation is in standard form: ( ax^2 + bx + c = 0 ).
• Identify the coefficients ( a ), ( b ), and ( c ).
  • Example: For ( 2x^2 + 8x + 6 = 0 ), ( a = 2 ), ( b = 8 ), ( c = 6 ).

Step 2: Factor the Equation

• Look for two numbers that multiply to ( ac ) (the product of ( a ) and ( c )) and add up to ( b ).
  • Example: In ( 2x^2 + 8x + 6 ):
    • ( ac = 2 \times 6 = 12 )
    • Find two numbers that multiply to 12 and add to 8. The numbers are 6 and 2.
• Rewrite the middle term using the two numbers found:
  • ( 2x^2 + 6x + 2x + 6 ).

Step 3: Group the Terms

• Group the terms into two pairs:
  • ( (2x^2 + 6x) + (2x + 6) ).
• Factor out the common factors from each group:
  • From the first group: ( 2x(x + 3) )
  • From the second group: ( 2(x + 3) ).

Step 4: Complete the Factoring

• Combine the factored terms:
  • ( 2x(x + 3) + 2(x + 3) )
  • This can be written as ( (x + 3)(2x + 2) ).
• Simplify the equation further if needed:
  • Factor out the common term: ( (x + 3)(2(x + 1)) ).

Step 5: Solve for the Roots

• Set each factor equal to zero:
  • ( x + 3 = 0 ) or ( 2(x + 1) = 0 ).
• Solve each equation:
  • From ( x + 3 = 0 ), we find ( x = -3 ).
  • From ( 2(x + 1) = 0 ), we find ( x + 1 = 0 ), leading to ( x = -1 ).
• The roots of the equation ( 2x^2 + 8x + 6 = 0 ) are ( x = -3 ) and ( x = -1 ).

Conclusion

Factoring is a powerful method for solving quadratic equations. By identifying the coefficients, factoring the equation, grouping terms, and solving for roots, you can efficiently find solutions to quadratic problems. Practice this method with various equations to enhance your proficiency. For further practice, consider solving different quadratic equations using the same steps outlined in this tutorial.