Problem 5 Solve each equation. $$u^{2}-3... [FREE SOLUTION]

Chapter 9: Problem 5

Solve each equation. $$u^{2}-300=0$$

Short Answer

Expert verified

u = \pm 10 \sqrt{3}

Step by step solution

Isolate the squared term

Start by isolating the squared term on one side of the equation. Add 300 to both sides of the equation. \[u^{2} - 300 = 0\] Adds 300: \[u^{2} = 300\]

Solve for the variable

Next, solve for the variable by taking the square root of both sides of the equation. Remember to include both the positive and negative roots. \[u = \sqrt{300}\] \[u = \pm \sqrt{300}\]

Simplify the square root

Simplify the square root of 300. Note that 300 can be written as $ 100 \times 3 $, so the square root of 300 can be simplified as follows: \[u = \pm \sqrt{100 \times 3}\] \[u = \pm (10 \sqrt{3})\]

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Key Concepts

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

square root

The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3, because 3 multiplied by 3 equals 9. When you take the square root of a number, you often see it represented by the symbol $ \sqrt{} $.
To solve a quadratic equation like $ u^{2} = 300 $, you need to take the square root of both sides. This operation helps isolate the variable (in this case, $ u $), making it easier to solve the equation.

simplifying radicals

Simplifying radicals is the process of breaking down a radical expression into its simplest form. For instance, to simplify $ \sqrt{300} $, start by looking for perfect square factors of 300.
Notice that 300 can be written as $ 100 \times 3 $. Because 100 is a perfect square, you can simplify:
\[ \sqrt{300} = \sqrt{100 \times 3} = \sqrt{100} \times \sqrt{3} = 10 \sqrt{3} \].
Here, 10 is the square root of 100, and $ \sqrt{3} $ remains under the radical because 3 is not a perfect square. Simplifying radicals makes it easier to handle and interpret mathematical expressions.

positive and negative roots

When you deal with square roots in equations, it's essential to remember that every positive number has two square roots: one positive and one negative. For example, the square roots of 300 are $ \sqrt{300} $ and $ -\sqrt{300} $.
This is because both $ 10 \sqrt{3} $ and $ -10 \sqrt{3} $ squared will give 300. In the context of solving quadratic equations like $ u^2 = 300 $, you need to consider both solutions: $ u = 10 \sqrt{3} $ and $ u = -10 \sqrt{3} $. Ignoring one of these roots would mean missing a part of the solution set.

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91影视

Solve each equation. $$u^{2}-300=0$$

Short Answer

Step by step solution

Isolate the squared term

Solve for the variable

Simplify the square root

Key Concepts

square root

simplifying radicals

positive and negative roots

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