How to Solve a Quadratic Equation by Synthetic Division?

Synthetic division is generally not used to solve a standard quadratic equation of the form ax² + bx + c = 0 directly. Instead, it's primarily employed to divide a polynomial by a linear factor of the form (x - k). The reference provided focuses on polynomial division, not solving quadratic equations specifically. However, if you know a root of a quadratic equation, you can use synthetic division to factor it. Here's how, along with a clarification of the method's typical use:

When Synthetic Division Can Help with Quadratics

Synthetic division shines when you are trying to factor a higher-degree polynomial, and you have already found one of its roots (perhaps by inspection, the rational root theorem, or using technology). In the context of a quadratic, this might be useful, though less common.

Example: Factoring a Quadratic Using a Known Root

Let's say we have the quadratic equation x² - 5x + 6 = 0, and we know that x = 2 is a root. We can use synthetic division to factor out (x - 2).

1. Set up the synthetic division:

   Write the coefficients of the quadratic (1, -5, and 6) and the known root (2) as follows:

   2 |  1  -5   6
       ----------------

2. Perform the synthetic division:

   • Bring down the first coefficient (1):

     2 |  1  -5   6
     ----------------
        1

   • Multiply the root (2) by the number you just brought down (1), and write the result (2) under the next coefficient (-5):

     2 |  1  -5   6
     ----------------
        1   2

   • Add the numbers in the second column (-5 + 2 = -3):

     2 |  1  -5   6
     ----------------
        1  -3

   • Multiply the root (2) by the result (-3), and write the result (-6) under the next coefficient (6):

     2 |  1  -5   6
     ----------------
        1  -3  -6

   • Add the numbers in the third column (6 + (-6) = 0):

     2 |  1  -5   6
     ----------------
        1  -3   0

3. Interpret the results:

   • The last number (0) is the remainder. Since it's 0, it confirms that (x - 2) is indeed a factor.
   • The other numbers (1 and -3) are the coefficients of the quotient. In this case, the quotient is x - 3.

4. Write the factored form:

   The original quadratic equation x² - 5x + 6 = 0 can now be written as (x - 2)(x - 3) = 0.

5. Solve for x:

   Setting each factor to zero, we get x - 2 = 0 => x = 2 and x - 3 = 0 => x = 3.

The More Common Method: Factoring, Quadratic Formula, or Completing the Square

For directly solving a quadratic equation, the following methods are far more common and efficient than relying on synthetic division:

• Factoring: Try to factor the quadratic expression into two binomials (as shown above after using synthetic division to find one root).
• Quadratic Formula: Use the quadratic formula: x = (-b ± √(b² - 4ac)) / 2a
• Completing the Square: Manipulate the equation to form a perfect square trinomial.

Conclusion

While you can technically use synthetic division to factor a quadratic if you already know one of its roots, it's not the standard or most efficient method for solving quadratic equations from scratch. Factoring, the quadratic formula, and completing the square are generally preferred. Synthetic division is more useful for simplifying higher-degree polynomials when a root is known or suspected.