Problem 38
Question
\(-3 m^{2}=-60\)
Step-by-Step Solution
Verified Answer
m = \pm 2\sqrt{5}
1Step 1: Isolate the variable term
To isolate the term with the variable, divide both sides of the equation by the coefficient of the variable term. In this case, divide by -3: \[-3 m^{2} \div -3 = -60 \div -3\]
2Step 2: Simplify both sides
Simplify the equation by performing the division: \[m^{2} = 20\]
3Step 3: Take the square root of both sides
To solve for \(m\), take the square root of both sides of the equation. Remember to consider both the positive and negative roots: \[m = \pm \sqrt{20}\]
4Step 4: Simplify the square root
Simplify \(\sqrt{20}\). Since \(20 = 4 \times 5\), \(\sqrt{20} = \sqrt{4 \times 5} = \sqrt{4} \times \sqrt{5} = 2\sqrt{5}\). So, \[m = \pm 2\sqrt{5}\]
Key Concepts
Isolating VariablesSquare RootsSimplification of Radicals
Isolating Variables
In mathematical equations, isolating the variable is key to finding its value. This means getting the variable by itself on one side of the equation. Imagine you have an equation \(-3 m^{2}=-60\). Here, the term \(-3m^{2}\) includes our variable. \(-60\) is a constant.
To isolate \(m^{2}\), we need to remove the coefficient -3. How do we do this? By performing the same operation on both sides of the equation. Divide both sides by -3: \[-3 m^{2} \div -3 = -60 \div -3\]. This simplifies to \(m^{2} = 20\).
Now, the variable term \(m^{2}\) is isolated. We have achieved our goal. Isolating variables is a critical first step in solving many algebraic equations.
To isolate \(m^{2}\), we need to remove the coefficient -3. How do we do this? By performing the same operation on both sides of the equation. Divide both sides by -3: \[-3 m^{2} \div -3 = -60 \div -3\]. This simplifies to \(m^{2} = 20\).
Now, the variable term \(m^{2}\) is isolated. We have achieved our goal. Isolating variables is a critical first step in solving many algebraic equations.
Square Roots
Once the variable is isolated, the next step often involves finding the value of the variable. In our example, we end up with \(m^{2} = 20\). To solve for \(m\), we take the square root of both sides.
Square roots are the opposite of squaring a number. When we take the square root of \(m^{2}\), we get \(m\). But remember, both the positive and negative roots must be considered. So, \[m = \pm \sqrt{20}\].
In summary:
Square roots are the opposite of squaring a number. When we take the square root of \(m^{2}\), we get \(m\). But remember, both the positive and negative roots must be considered. So, \[m = \pm \sqrt{20}\].
In summary:
- Isolate the variable term \(m^{2}\).
- Take square roots of both sides, considering \(\pm\) roots.
- Simplify the result, if necessary.
Simplification of Radicals
Radicals, or square roots, often need simplification. Simplifying radicals means making them as simple as possible. In our example, we reached \(m = \pm \sqrt{20}\).
We simplify \(\sqrt{20}\) by finding its prime factors. Since \(20 = 4 \times 5\), we can rewrite \(\sqrt{20}\) as \(\sqrt{4\times5}\). Using the property \(\sqrt{a\times b} = \sqrt{a}\times\sqrt{b}\), we get \(\sqrt{4}\times\sqrt{5} = 2\sqrt{5}\).
Hence, simplifying \(\sqrt{20}\) gives us \(2\sqrt{5}\). Thus, \[m = \pm 2\sqrt{5}\]. To summarize:
We simplify \(\sqrt{20}\) by finding its prime factors. Since \(20 = 4 \times 5\), we can rewrite \(\sqrt{20}\) as \(\sqrt{4\times5}\). Using the property \(\sqrt{a\times b} = \sqrt{a}\times\sqrt{b}\), we get \(\sqrt{4}\times\sqrt{5} = 2\sqrt{5}\).
Hence, simplifying \(\sqrt{20}\) gives us \(2\sqrt{5}\). Thus, \[m = \pm 2\sqrt{5}\]. To summarize:
- Break down the number under the radical into prime factors.
- Use properties of square roots to simplify.
- Write the simplified radical.