Problem 98 - College Algebra

Question

Solve each equation in Exercises 73-98 by the method of your choice. $$\frac{3}{x-3}+\frac{5}{x-4}=\frac{x^{2}-20}{x^{2}-7 x+12}$$

Step-by-Step Solution

Answer

The given equation has no solution in the real number set.

Step 1: Simplification

Rewrite the equation \(\frac{3}{x-3}+\frac{5}{x-4}=\frac{x^{2}-20}{x^{2}-7 x+12}\) in the form \(\frac{A}{x-3}+\frac{B}{x-4}=\frac{C}{(x-3)(x-4)}\), where \(A\), \(B\), and \(C\) are expressions to be determined. Multiply throughout by \((x-3)(x-4)\) to clear the fractions. This gives you \(3(x-4)+5(x-3)=x^{2}-20\). After simplification, this becomes \(8x-27=x^{2}-20\).

Step 2: Solving the quadratic equation

Rearrange the equation from Step 1 and set it to zero: \(x^{2}-8x+47=0\). Now the equation becomes a quadratic equation, and it can be solved by the quadratic formula \(x=\frac{-b±\sqrt{b^2-4ac}}{2a}\). The coefficients from our equation are \(a=1\), \(b=-8\), and \(c=47\). Substituting these values into the formula gives the roots of the equation as \(x_{1,2}=\frac{8±\sqrt{(-8)^{2}-4*1*47}}{2*1}\).

Step 3: Calculation and verification

The numbers under the square root in the formula \(x_{1,2}\) from Step 2 are imaginary numbers because the discriminant \(b^{2}-4ac<0\). Since we are dealing with real numbers, there will be no solutions for the equation.

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations, which are equations of the form \( ax^2 + bx + c = 0 \). It provides a straightforward way to find the 'roots' or solutions of the quadratic equation. The formula is:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

Here:

• \(a\), \(b\), and \(c\) are coefficients of the terms in the equation.
• \(\pm\) indicates two solutions: one for adding and one for subtracting the square root.
• The term under the square root, \( b^2 - 4ac \), is called the "discriminant" and plays a crucial role in determining the nature of the roots.

This formula is particularly useful when the quadratic equation does not factor easily. Multiplying through by the common denominator and rearranging terms to create a standard quadratic form is the first step in using the quadratic formula.

Discriminant

The discriminant of a quadratic equation \( ax^2 + bx + c = 0 \) is the part \( b^2 - 4ac \) found under the square root in the quadratic formula. It provides critical information about the nature of the roots of the equation:

• If the discriminant is positive, \(b^2 - 4ac > 0\), the quadratic equation has two distinct real roots.
• If it is zero, \(b^2 - 4ac = 0\), there is exactly one real root, also known as a repeated or double root.
• If the discriminant is negative, \(b^2 - 4ac < 0\), the equation has no real solutions, but rather two complex conjugate roots, involving imaginary numbers.

Understanding the discriminant is vital since it tells us beforehand whether we should expect real or imaginary results from solving the equation. In the case of our exercise, the discriminant is negative, leading us to solutions that involve imaginary numbers.

Imaginary Numbers

Imaginary numbers are a fascinating component of mathematics that come into play when we encounter square roots of negative numbers. Typically, the square root of a negative number cannot be computed within the realm of real numbers. Therefore, we introduce the imaginary unit, denoted by \( i \), where \( i = \sqrt{-1} \).

• This allows us to express roots of negative numbers. For example, \( \sqrt{-4} = 2i \).
• Once we include imaginary numbers, we expand our number system to include complex numbers, which have a 'real' part and an 'imaginary' part. A typical complex number looks like \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part.

Imaginary numbers are essential in various fields like engineering and physics, as they provide solutions to otherwise unsolvable problems. In our equation, since the discriminant is negative, the roots are imaginary, indicating there are no "real" solutions to the problem.

Simplification of Rational Expressions

Simplification of rational expressions is a crucial skill in algebra that makes handling fractions with polynomials easier and more manageable. A rational expression is a fraction in which both the numerator and the denominator are polynomials. Simplifying these involves:

• Finding and eliminating common factors from the numerator and the denominator.
• Rewriting complex fractions in simpler terms, usually through a common denominator or polynomial division.

In the initial form of the equation from the exercise, simplification involved rewriting each term so the mathematical operations could clear the fractions and transform it into a quadratic equation. Multiply by the least common denominator when you need to eliminate fractional terms fully. In this problem, breaking down the equation to free it from its rational expression led to an easier resolution using the quadratic formula.