Problem 43 - Algebra 1: Concepts and Skills

Question

Solve \(\frac{c}{c-4}=\frac{8}{c-10}\). A. 0 B. 2 and 16 C. -18 and 32 D. No solution

Step-by-Step Solution

Answer

The solutions to the equation are \(c = 2, 16\), so the correct answer is 'B. 2 and 16'.

Step 1: Cross multiply

Cross multiply to get rid of the fractions: \(c \cdot (c - 10) = 8 \cdot (c - 4)\)

Step 2: Apply distributive property

Apply the distributive property (multiplying each term of the binomial to the \(c\) or 8 outside the brackets): \(c^2 -10c = 8c -32\)

Step 3: Form a quadratic equation

Form a quadratic equation by bringing all the terms to one side of the equation: \(c^2 - 10c - 8c + 32 = 0\). This simplifies to: \(c^2 - 18c + 32 = 0\)

Step 4: Apply the quadratic formula

Solve the quadratic equation using the quadratic formula \(c = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\), \(b = -18\), and \(c = 32\). This process gives \(c = 2, 16\)

Key Concepts

Cross Multiplication

Cross multiplication is a technique used to simplify equations that involve fractions. It helps us clear the fractions and work with whole numbers instead. In a rational equation like \(\frac{c}{c-4}=\frac{8}{c-10}\), the first step is to cross multiply.
Imagine drawing an 'X' diagonally across the fractions, which indicates the two products you will create: \(c \times (c - 10)\) on one side and \(8 \times (c - 4)\) on the other side. Effectively, what you're doing is multiplying the numerator of one fraction by the denominator of the other fraction, and setting these products equal to each other:
\[c(c-10) = 8(c-4)\]
This process eliminates the fractions, allowing us to focus on solving an equation without them.

Distributive Property

After cross-multiplying, the equation \(c(c-10) = 8(c-4)\) can be further simplified using the distributive property. The distributive property allows us to multiply a single term by each term inside a bracket.
Here, for instance, on the left side, we distribute \(c\) to both \(c\) and \(-10\), giving us \(c^2 - 10c\). Similarly, on the right side, distribute \(8\) to both \(c\) and \(-4\), which results in \(8c - 32\).
Combining these, the equation becomes:

• \(c^2 - 10c = 8c - 32\)

By using the distributive property, we convert complex expressions into simpler ones that are more straightforward to manage.

Quadratic Equation

After using the distributive property, we have the expression \(c^2 - 10c = 8c - 32\). The next step is to form a quadratic equation.
A quadratic equation is one that can be written in the form \(ax^2 + bx + c = 0\). To achieve this form, we need to move all terms to one side.
Hence, subtract \(8c\) from both sides and add \(32\) to both sides:

• \(c^2 - 18c + 32 = 0\)

This equation is quadratic because the highest exponent of \(c\), the variable, is 2. Solving such equations requires specific methods like factoring, completing the square, or using the quadratic formula.

Quadratic Formula

The quadratic formula is a reliable method to solve any quadratic equation of the form \(ax^2 + bx + c = 0\). It is expressed as:
\[c = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
In our equation \(c^2 - 18c + 32 = 0\), the coefficients are \(a = 1\), \(b = -18\), and \(c = 32\). By substituting these into the quadratic formula, we can solve for values of \(c\).
First, compute the discriminant \(b^2 - 4ac\), which determines the nature of the roots. It turns out:

• \((-18)^2 - 4 \times 1 \times 32 = 324 - 128 = 196\)

Since the discriminant is positive, it suggests there are two distinct real roots.
Plug the values into the formula:

• \(c = \frac{-(-18) \pm \sqrt{196}}{2 \times 1} = \frac{18 \pm 14}{2}\)
• Simplifying, gives \(c = 16\) and \(c = 2\)

These solutions satisfy the original equation, making \(c = 2\) and \(c = 16\) the correct solutions.