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Solving Quadratic Equations by Factoring

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Solving Quadratic Equations by Factoring

Quadratic equations are equations that contain a term with the variable squared (x2). They often appear in various fields, including physics, engineering, and economics. Solving quadratic equations is a fundamental skill in algebra, and one common method is factoring.

Understanding Quadratic Equations

A quadratic equation is typically written in the standard form:

ax2 + bx + c = 0

where a, b, and c are constants, and a ≠ 0. The goal is to find the values of x that satisfy the equation.

Steps to Solve Quadratic Equations by Factoring

  1. Put the equation in standard form: Ensure the equation is arranged with all terms on one side and set equal to zero.
  2. Factor the quadratic expression: Find two binomials that multiply to give the original quadratic expression. This involves finding two numbers that add up to ‘b’ and multiply to ‘c’.
  3. Set each factor equal to zero: Once you have factored the expression, set each factor equal to zero and solve for x.

Example

Let’s solve the quadratic equation: x2 + 5x + 6 = 0

  1. Standard form: The equation is already in standard form.
  2. Factoring: Find two numbers that add up to 5 (the coefficient of x) and multiply to 6 (the constant term). These numbers are 2 and 3. So, we can factor the expression as: (x + 2)(x + 3) = 0
  3. Setting factors to zero:
    • x + 2 = 0 => x = -2
    • x + 3 = 0 => x = -3

Therefore, the solutions to the quadratic equation x2 + 5x + 6 = 0 are x = -2 and x = -3.

Important Notes

  • Avoid dividing by x: Never divide both sides of the equation by x, as this could lead to the loss of a solution (x = 0).
  • Check your solutions: Always substitute the solutions back into the original equation to verify that they are correct.

Additional Resources

For more practice and detailed explanations on factoring quadratic equations, you can refer to these resources:

Conclusion

Solving quadratic equations by factoring is a powerful technique that allows you to find the roots of a quadratic expression. By understanding the steps involved and practicing with examples, you can master this essential algebraic skill.

Factoring