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Chapter 1: Problem 27

Find the real solutions, if any, of each equation. $$ \left|x^{2}-9\right|=0 $$

Short Answer

Expert verified
The real solutions are \( x = 3 \) and \( x = -3 \).

Step by step solution

01

Understand the Absolute Value

The given equation is \ \ \( \left|x^{2}-9\right|=0 \). Absolute value equations can be rewritten without the absolute value if the expression inside is equal to zero, because the absolute value of zero is zero.
02

Set the Inside Equal to Zero

Set the expression inside the absolute value equal to zero: \ \ \( x^2 - 9 = 0 \)
03

Solve the Quadratic Equation

To solve \( x^2 - 9 = 0 \), add 9 to both sides of the equation: \ \ \( x^2 = 9 \). Then, take the square root of both sides: \ \ \( x = \pm 3 \).
04

Verify the Solutions

Substitute \( x = 3 \) and \( x = -3 \) back into the original equation to verify: \ \ \( \left|3^2 - 9\right| = \left|9 - 9\right| = \left|0\right| = 0 \). Similar verification shows \( x = -3 \) also satisfies the equation.

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Key Concepts

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

solving quadratic equations
When you encounter a quadratic equation, you need to find the values of the variable that make the equation true. Quadratic equations are typically in the form \[ax^2 + bx + c = 0\]. For our specific exercise, we have \[x^2 - 9 = 0\]. Here, the task is to solve for \(x\). To do this, follow these steps:
  • Add 9 to both sides: \[x^2 = 9\].
  • Take the square root of both sides: \[x = \pm 3\]. The \(\pm\) symbol means there are two solutions: 3 and -3.
Quadratic equations can often be solved by factoring, completing the square, or using the quadratic formula. But for this example, recognizing it as a simple difference of squares was key.
absolute value properties
The absolute value of a number is its distance from zero on the number line, regardless of direction. For an equation like \[|x^2 - 9| = 0\], this means the expression inside the absolute value brackets must equal zero because \[|0| = 0\]. Thus, you can rewrite the equation without the absolute value when the context of the problem confirms the inside is zero:
  • Set the inside equal to zero: \[x^2 - 9 = 0\].
  • From here, solve the quadratic equation within, as demonstrated in the previous section.
It's essential to remember absolute value properties because they simplify solving such equations.
verifying solutions
After solving an equation, verifying your solutions ensures they satisfy the original problem. For our exercise, after finding \(x = \pm 3\), substitute these solutions back into the original absolute value equation to check:
  • For \(x = 3\): \[|3^2 - 9| = |9 - 9| = |0| = 0\].
  • For \(x = -3\): \[|-3^2 - 9| = |9 - 9| = |0| = 0\].
Both values satisfy the original equation, confirming they are correct solutions. Always perform this step to catch potential errors and validate your answers.

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