Solving quadratic equation by factoring

General
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solving quadratic equation by factoring

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Solving Quadratic Equations by Factoring

Answer:
Solving a quadratic equation by factoring is a method that involves expressing the quadratic equation in the form of a product of binomials set to zero. Here’s a step-by-step approach to solve quadratic equations by factoring:

Solution By Steps:

  1. Write the Quadratic Equation in Standard Form:

    • Ensure that the quadratic equation is in the standard form ax^2 + bx + c = 0.

  2. Factor the Quadratic Expression:

    • Find two binomials whose product gives the quadratic expression ax^2 + bx + c.

  3. Set Each Factor Equal to Zero:

    • Apply the Zero Product Property, which states that if the product of two factors equals zero, then at least one of the factors must be zero.

  4. Solve for x:

    • Solve the resulting linear equations to find the values of x.

Example Problem:

Solve the quadratic equation by factoring: ( x^2 - 5x + 6 = 0 )

  1. Write the Quadratic Equation in Standard Form:

    • The quadratic equation is already in standard form ( x^2 - 5x + 6 = 0 ).

  2. Factor the Quadratic Expression:

    • Look for two numbers that multiply to give the constant term (6) and add to give the coefficient of the linear term (-5). These numbers are -2 and -3.

      x^2 - 5x + 6 = (x - 2)(x - 3)

  3. Set Each Factor Equal to Zero:

    • Use the Zero Product Property.

      x - 2 = 0 \quad \text{or} \quad x - 3 = 0

  4. Solve for x:

    • Solve each linear equation for ( x ).

      x - 2 = 0 \implies x = 2
      x - 3 = 0 \implies x = 3

Final Answer:
The solutions to the quadratic equation ( x^2 - 5x + 6 = 0 ) are ( x = 2 ) and ( x = 3 ).

Another Example:

Solve the quadratic equation by factoring: ( x^2 + 4x - 12 = 0 )

  1. Write the Quadratic Equation in Standard Form:

    • The quadratic equation is already in standard form ( x^2 + 4x - 12 = 0 ).

  2. Factor the Quadratic Expression:

    • Look for two numbers that multiply to give the constant term (-12) and add to give the coefficient of the linear term (4). These numbers are 6 and -2.

      x^2 + 4x - 12 = (x + 6)(x - 2)

  3. Set Each Factor Equal to Zero:

    • Use the Zero Product Property.

      x + 6 = 0 \quad \text{or} \quad x - 2 = 0

  4. Solve for x:

    • Solve each linear equation for ( x ).

      x + 6 = 0 \implies x = -6
      x - 2 = 0 \implies x = 2

Final Answer:
The solutions to the quadratic equation ( x^2 + 4x - 12 = 0 ) are ( x = -6 ) and ( x = 2 ).

Solving quadratic equations by factoring is a useful technique when the equation can be factored easily. It is efficient and provides the exact values of the solutions.