Problem 5 Solve for x. $$ x^{2}-100=0 ... [FREE SOLUTION]

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Painless Algebra

Lynette Long

$Math Studyset 91影视 Explanations$ Math

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Chapter 9: Problem 5

Solve for x. $$ x^{2}-100=0 $$

Short Answer

Expert verified

x = 10 or x = -10

Step by step solution

Identify the Equation

The given equation is a quadratic equation: \[ x^{2} - 100 = 0 \]

Isolate the Squared Term

Add 100 to both sides of the equation to isolate the squared term:\[ x^{2} - 100 + 100 = 0 + 100 \] Which simplifies to: \[ x^{2} = 100 \]

Solve for x

Take the square root of both sides of the equation to solve for x:\[ \sqrt{x^2} = \sqrt{100} \] Remember that the square root of a number has both a positive and a negative solution:\[ x = \pm 10 \]

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Key Concepts

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

Quadratic Equations

Quadratic equations are a common type of algebraic equation that take the form $ ax^{2} + bx + c = 0 $, where 'a,' 'b,' and 'c' are constants. In this problem, we see a simplified quadratic equation, $ x^{2} - 100 = 0 $. These equations often have two solutions and can be solved using various methods such as factoring, completing the square, or using the quadratic formula.
Here, we observed one method of solving a simplified quadratic equation. By first isolating $ x^{2} $, we can simplify the process and find the solutions more easily.

Square Roots

The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 100 is 10, because $ 10 \times 10 = 100 $. When solving quadratic equations like $ x^{2} = 100 $, we solve for 'x' by taking the square root of both sides.
However, it's important to remember that square roots have both positive and negative solutions. In this problem, taking the square root of 100 gives us both $ +10 $ and $ -10 $. This is why solving $ x^{2} = 100 $ provides us with two possible values for 'x' 鈥� one positive and one negative.

Positive and Negative Solutions

Quadratic equations often yield two solutions because the square of either a positive or a negative number will give the same result. For instance, both $ 10^{2} $ and $ (-10)^{2} $ equal 100. This principle ensures that a quadratic equation like $ x^{2} = 100 $ has two possible solutions: $ x = 10 $ and $ x = -10 $.
This dual nature of solutions is crucial to understand, especially when solving real-world problems, as it can affect the context in which the solutions apply. For example, in some cases, only the positive solution may be meaningful (e.g., distance or length), while in other scenarios, both might be relevant.

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91影视

Solve for x. $$ x^{2}-100=0 $$

Short Answer

Step by step solution

Identify the Equation

Isolate the Squared Term

Solve for x

Key Concepts

Quadratic Equations

Square Roots

Positive and Negative Solutions

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