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Chapter 12: Problem 92

Solve the equation. $$x^{2}-30=-3$$

Short Answer

Expert verified
The roots of the quadratic equation \(x^{2}-30=-3\) are \(x = 3\sqrt{3}\) and \(x = -3\sqrt{3}\).

Step by step solution

01

Move All Terms to One Side of the Equation

First, balance the equation by moving the constant on the right side to the left side, resulting in \(x^{2}-30+3=0\).
02

Simplify the Equation

Simplify the left side of the equation to get a clearer expression. \(x^{2}-27=0\).
03

Solve the Simplified Equation for \(x\)

Apply the square root property (\(a^2 = b \Rightarrow a = \sqrt{b}\)) to both sides to solve for \(x\). This gives us two roots: \(x = \sqrt{27}\) and \(x = -\sqrt{27}\). We simplify to get: \(x = 3\sqrt{3}\) and \(x = -3\sqrt{3}\).

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Key Concepts

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

Solving Equations
Solving equations is all about finding the value of the variable, often represented as \(x\), that makes an equation true. In this particular problem, we start with the equation \(x^2 - 30 = -3\). The first step in solving such equations is to get all terms involving \(x\) on one side and constants on the other. Here鈥檚 a simple guideline:
  • Rearrange the equation to isolate the variable term: \(x^2 - 30 + 3 = 0\).
  • Combine like terms to simplify: \(x^2 - 27 = 0\).
Once this is done, the path to finding \(x\) becomes clearer. This strategy helps break down complex problems into manageable steps, making it easier to find the solution.
Square Root Property
The square root property is a useful tool for solving quadratic equations when you have an expression of the form \(x^2 = c\). It states that if \(a^2 = b\), then \(a\) equals \(\sqrt{b}\) or \(-\sqrt{b}\). In our example, once simplified, the equation becomes \(x^2 = 27\). Here is how you use the square root property:
  • Take the square root of both sides: \(x = \sqrt{27}\) or \(x = -\sqrt{27}\).
  • These square roots give you the potential solutions for \(x\).
It's vital to remember that when you take the square root of both sides of an equation, you account for both the positive and negative roots. This step is crucial as every square has two roots.
Simplifying Expressions
Simplifying expressions is an essential part of solving equations that helps in making the problem easier to handle. In this problem, after applying the square root property, we get expressions like \(x = \sqrt{27}\) and \(x = -\sqrt{27}\). To further simplify:
  • Break down the number under the square root: \(\sqrt{27} = \sqrt{9 \times 3} = \sqrt{9} \times \sqrt{3}\).
  • Simplify the square roots: \(\sqrt{9} = 3\), thus \(x = 3\sqrt{3}\) and \(x = -3\sqrt{3}\).
This reduction makes it easy to interpret the solution, providing a clearer and more straightforward expression. Simplifying expressions is not just about making the numbers prettier; it鈥檚 about revealing the core values that lie beneath the complex appearance of the equation.

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