Solving Quadratic Equations with Integer Solutions

When we encounter a quadratic equation whose solutions are integers, we can often solve it by factoring. This is a powerful technique that can simplify the problem significantly. Let's explore the reasons and methods behind this approach, and provide a few examples to illustrate the process.

When Can We Factor the Equation?

If integers m and n are the solutions to a quadratic equation, the equation can be factored in the form of ax^2 - mx - n 0. This is a key observation that makes factoring an effective method for solving certain quadratic equations.

Case Study: Solving a Quadratic Equation with Integers

Consider the quadratic equation x^2 - 83x - 630 0. At first glance, it might seem challenging to factor. However, if we use the quadratic formula to find the roots, we can sometimes spot that the solutions are nearly integers. In this case, the quadratic formula gives the roots as 83 - sqrt(9409)/2. Upon closer inspection, we can observe that 9409 is a perfect square, and sqrt(9409) is approximately 97. Using this insight, we can rewrite the equation as (x 7)(x - 90) 0, leading to the roots -7 and 90. Here's the step-by-step process:

Use the quadratic formula to find the roots. Notice that the solutions are very close to integers. Verify that the square root is indeed an integer. Factor the equation using the identified integers. Write the solution clearly on a clean sheet of paper.

Another Example: Solving by Factoring

Consider the equation x^2 - 3x - 2 0. To factor this, we need to find two integers whose product is -2 and whose sum is -3. The integers 2 and -1 satisfy these conditions, so we can factor the equation as (x - 2)(x 1) 0, leading to the roots x 2 and x -1.

Example of Direct Factoring

Let's look at a simpler example: x^2 - 5x - 6 0. We need to find two integers whose product is -6 and whose sum is -5. The integers -2 and -3 fit this description, so we can factor the equation as (x 2)(x - 3) 0, giving us the roots x -2 and x 3.

Conclusion

When the solutions to a quadratic equation are integers, it is often possible to solve the equation by factoring. This method is straightforward and efficient. However, not all quadratic equations can be solved so easily. For instance, the sum of cubes problem for 42 involves non-integer solutions and cannot be solved by factoring.

Relating to the Quadratic Formula

Using the quadratic formula, x [-b ± sqrt(b^2 - 4ac)] / (2a), we can find the roots of any quadratic equation. For equations with integer roots, we can often spot these roots by inspection or by recognizing when the discriminant (b^2 - 4ac) is a perfect square.

Additional Resources

For more information on solving quadratic equations, you can refer to the following resources:

Quadratic Equation on Wikipedia Intermediate Algebra: Quadratic Equations (Solving by Factoring)