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Chapter 9: Problem 17

Solve each equation. $$\frac{2}{5} a^{2}+3=11$$

Short Answer

Expert verified
The solutions are \( a = 2\sqrt{5} \) and \( a = -2\sqrt{5} \).

Step by step solution

01

- Isolate the quadratic term

Start by moving the constant term 3 to the other side of the equation. Subtract 3 from both sides:\[\frac{2}{5} a^2 + 3 - 3 = 11 - 3\]This simplifies to:\[\frac{2}{5} a^2 = 8\]
02

- Clear the fraction

Multiply both sides of the equation by the reciprocal of \( \frac{2}{5} \), which is \( \frac{5}{2} \). This will eliminate the fraction on the left side:\[\left( \frac{5}{2} \right) \frac{2}{5} a^2 = 8 \left( \frac{5}{2} \right)\]This simplifies to:\[a^2 = 20\]
03

- Solve for a

Take the square root of both sides to solve for \( a \). Remember to consider both the positive and negative solutions:\[a = \pm \sqrt{20}\]Simplify the square root:\[a = \pm \sqrt{4 \cdot 5} = \pm 2 \sqrt{5}\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isolating the Variable
First, it's important to understand what isolating the variable means. It's a technique where we rearrange the equation so that the variable we are solving for stands alone on one side of the equation. Let's look at our example: \(\frac{2}{5} a^{2} + 3 = 11\). The goal is to get \(a^{2}\) by itself.

We start by getting rid of the constant term 3 on the left side. To do this, we subtract 3 from both sides of the equation:

\(\frac{2}{5} a^{2} + 3 - 3 = 11 - 3\)

Now the equation simplifies to:

\(\frac{2}{5} a^{2} = 8\)

Now, the variable term \(a^{2}\) is isolated. This step makes it easier to solve the next steps.
Clearing Fractions
Now let's move into clearing fractions. When an equation involves fractions, it's often easiest to eliminate them to simplify the equation.

In our example, we have a fraction \(\frac{2}{5} a^{2}\). To get rid of this fraction, we multiply both sides of the equation by its reciprocal. The reciprocal of \(\frac{2}{5}\) is \(\frac{5}{2}\):

\(\frac{5}{2} \times \frac{2}{5} a^{2} = 8 \times \frac{5}{2}\)

This operation simplifies the equation to:

\(a^{2} = 20\)

The fraction has been effectively cleared, making it much easier to solve.
Taking Square Roots
Next, we need to solve for the variable by taking the square root of both sides of the equation.

We have:

\(a^{2} = 20\)

To isolate \(a\), take the square root of both sides. Remember to consider both the positive and negative roots when dealing with square roots:

\(a = \pm \sqrt{20}\)

This is because both \(4\) and \(-4\) squared give \(16\). The same logic applies here for the \(20\), resulting in both positive and negative solutions.
Simplifying Radicals
Finally, we simplify the radical expression \(\sqrt{20}\). Simplifying radicals makes the equation cleaner and typically easier to understand.

Break \(\sqrt{20}\) down into its factors:

\(\sqrt{20} = \sqrt{4 \cdot 5}\)

Since the square root of 4 is 2, we can simplify it to:

\(\sqrt{20} = 2 \sqrt{5}\)

Thus, the solution involves both the positive and negative values of this simplified form:

\(a = \pm 2 \sqrt{5}\).

So, isolating the variable, clearing fractions, taking square roots, and simplifying radicals work together to solve the quadratic equation.

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Most popular questions from this chapter

Graph each function using transformations. $$f(x)=(x-6)^{2}-2$$

(a) rewrite each function in \(f(x)=a(x-h)^{2}+k\) form and (b) graph it using properties. $$f(x)=-2 x^{2}-4 x-5$$

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