Problem 68 - Algebra for College Students

Question

Either solve the given equation or perform the indicated operation \((s),\) whichever is appropriate. $$\frac{5}{h}=\frac{h}{5}$$

Step-by-Step Solution

Answer

h = 5 or h = -5

Step 1: Cross Multiply

To solve the equation \(\frac{5}{h}=\frac{h}{5}\), start by cross-multiplying the terms to eliminate the fractions. This gives us: \[5 \times 5 = h \times h\] which simplifies to \[25 = h^2\]

Step 2: Solve for h

Now, solve for \(h\) by taking the square root of both sides of the equation \[ h^2 = 25 \]. This gives us two solutions because the square root of 25 can be positive or negative: \[h = \pm 5\]

Key Concepts

cross-multiplication

Cross-multiplication is a technique used to simplify equations that involve fractions. In the equation \(\frac{5}{h}=\frac{h}{5}\), cross-multiplication helps to eliminate the fractions by multiplying across the equal sign. This means you multiply the numerator of one fraction by the denominator of the other. So, you get:

• 5 \(\times\) 5 on one side
• and h \( \times \) h on the other.

This simplifies to \(25 = h^2\). This step makes the equation easier to solve, as you're left with a polynomial equation without fractions.

solving quadratic equations

Once you have \(25 = h^2\), the next step is to solve the quadratic equation. Quadratic equations are equations that can be written in the form \(ax^2 + bx + c = 0\). In our example, \(h^2 - 25 = 0\) is already in the simplified form where \(a = 1\), \(b = 0\), and \(c = -25\).
The simplest method to solve this equation is by taking the square root of both sides.

• Starting from \(h^2 = 25\)
• take the square root of both sides to get \(h = \sqrt{25}\)
• \(h = \pm 5\).

The \(\pm\) symbol indicates that both positive and negative roots are possible solutions to the equation.

positive and negative solutions

When solving quadratic equations, it's important to consider both positive and negative solutions. In the equation \(h^2 = 25\), the solutions are found through taking the square root, which gives us \(h = \pm 5\). This means the equation has two solutions:

• \(h = 5\)
• and \(h = -5\).

Having both positive and negative solutions is common in quadratic equations. It is crucial to recognize and include both in your final answer to ensure that all possible solutions are considered. This dual nature of solutions reflects how functions can have multiple intersection points with the x-axis or represent different scenarios in real-world problems.

Always remember to verify these solutions in the original equation to confirm they satisfy it, which in this case they do.