Quadratic Equations with Integer Solutions: Solving through Factoring

Quadratic equations are ubiquitous in mathematics and their solutions can provide significant insight into various mathematical and real-world problems. One interesting aspect of quadratic equations involves cases where the solutions are integers. In such scenarios, the equation can often be factored, providing a straightforward method to find the roots. This article explores the conditions under which a quadratic equation with integer solutions can be solved by factoring, and provides examples to illustrate the process.

When Solutions Are Integers, Factoring Is Possible

Consider a standard form quadratic equation:

ax2 bx c 0

If the quadratic equation has integer solutions, the equation can indeed be factored into the product of two binomials with integer coefficients. This is possible because the integer solutions correspond to the values of x that make the equation equal to zero. Therefore, the quadratic can be expressed in the form:

ax - r1x - r2 0

where r1 and r2 are the integer roots of the equation.

However, it's crucial to note that not all quadratics with integer coefficients will necessarily have integer roots. The discriminant, given by:

D b2 - 4ac

must be a perfect square for the quadratic to have rational and, consequently, potentially integer solutions. If the discriminant is a perfect square, the roots can be calculated using the quadratic formula:

x frac{-b pm sqrt{D}}{2a}

If both roots are integers, then factoring is indeed a viable method to solve the equation.

Practical Example: Root Identification and Factoring

Consider the quadratic equation x2 - 83x - 630 0. Initially, one might attempt to factor it directly and find it challenging. Using the quadratic formula, we can determine the roots:

x frac{-(-83) pm sqrt{(-83)2 - 4(1)(-630)}}{2(1)} frac{83 pm sqrt{9409}}{2}

Upon further inspection, we observe that 9409 is a perfect square, specifically 972. This insight alerts us to the possibility that the square root of 9409 is 97 rather than its approximate value. Thus, the roots can be calculated as:

x frac{83 pm 97}{2}

which simplifies to:

x -7 and x 90

With these roots identified, we can now factor the quadratic as:

(x 7)(x - 90) 0

This method of solving the quadratic by recognizing the perfect square through an approximate calculation and then factoring the equation can be a powerful technique in algebra.

Factorization Process

Finding the factors of a quadratic equation when the solutions are integers can be simplified by utilizing a specific formula:

x2 - abx - ab 0

To factor the equation, one looks for two integers that multiply to ab and add up to ab. This process can be quite straightforward with simple values, such as in the example:

x2 - 3x 2 0

Here, the goal is to find two integers whose product is 2 and whose sum is 3. The numbers 2 and 1 satisfy both conditions:

2 * 1 2 2 1 3

Using these values, we can factor the quadratic as:

(x - 1)(x - 2) 0

This article has illustrated the process of solving quadratic equations with integer solutions through factoring. Understanding and applying these methods can greatly enhance one's problem-solving skills in algebra and beyond.