Problem 8 - Elementary Algebra

Question

Solve. $$ (h+2)(h-10)=0 $$

Step-by-Step Solution

Answer

The solutions are \ h = -2 \ and \ h = 10 \.

Step 1: Identify the Zero Product Property

The Zero Product Property states that if a product of two factors is zero, then at least one of the factors must be zero. Therefore, if \( (h+2)(h-10)=0 \), then either \(h + 2 = 0\) or \(h - 10 = 0\).

Step 2: Solve the First Equation

Start by solving \( h + 2 = 0 \). To isolate \( h \), subtract 2 from both sides: \ h + 2 - 2 = 0 - 2 \ Simplifying this, we get \ h = -2 \.

Step 3: Solve the Second Equation

Next, solve \( h - 10 = 0 \). To isolate \( h \), add 10 to both sides: \ h - 10 + 10 = 0 + 10 \ Simplifying this, we get \ h = 10 \.

Step 4: Combine Solutions

The solutions to the equation \( (h+2)(h-10)=0 \) are \ h = -2 \ and \ h = 10 \.

Key Concepts

solving quadratic equations

When we tackle quadratic equations like $$ (h+2)(h-10)=0 $$, we're dealing with an equation of the form $$ ax^2 + bx + c = 0 $$. Quadratic equations can shoot out a curve on a graph, typically creating a **parabola**. Our goal is to find the points where this curve slices through the x-axis. These points are called ***roots*** or ***solutions*** of the equation.

One clear way to solve quadratic equations is through **factoring**. Factoring involves expressing the equation as a product of two binomials. For example, in our equation, it’s already in factored form: $$ (h+2)(h-10)=0 $$.

The roots come from the Zero Product Property, which tells us that if the product of two expressions is zero, at least one must be zero. This simplifies our task tremendously! Instead of dealing with one complex equation, we get two simpler ones: $$h+2=0$$ and $$h-10=0$$. Solving these will give us our solutions.

factoring

Factoring is a **key skill** in solving quadratic equations. It involves breaking down an equation or expression into simpler parts, which when multiplied together will give you the original equation.

In this exercise, we are presented with $$ (h+2)(h-10)=0 $$. Fortunately, this equation is already factored. But if it weren't, we’d need to factor it ourselves. Here’s how to factor a common quadratic equation, step-by-step:

• Write the equation in standard form: $$ ax^2 + bx + c = 0 $$.
• Find two numbers that multiply to $$ a \times c $$ and add up to $$ b $$.
• Use these numbers to split the middle term and factor by grouping.
• Set each factor (binomial) to zero and solve for the variable.

In our specific example, $$ a = 1 $$, $$ b = -8 $$, and $$ c = -20 $$ aren’t needed steps as it’s already factored. Applying the Zero Product Property instantly shaves down our workload, making it a bit breezier!

isolating variables

Isolating variables is crucial for solving any algebraic equation. It means getting the variable by itself on one side of the equation.

In the exercise example, we have two simple linear equations after applying the Zero Product Property:

• $$ h+2=0 $$
• $$ h-10=0 $$

We isolate the variable in each:

• For $$ h+2=0 $$, we subtract 2 from both sides: $$ h = -2 $$.
• For $$ h-10=0 $$, we add 10 to both sides: $$ h = 10 $$.

Voilá! We’ve isolated $$ h $$ in both equations, giving us our solutions: $$ h=-2 $$ and $$ h=10 $$.

Always remember, isolating the variable is like peeling an onion—shed the extraneous layers step by step (by adding or subtracting terms) until you’re left with the core—the variable itself!