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Chapter 3: Problem 53

Solve each absolute value equation. $$\left|w^{2}-4\right|=3$$

Short Answer

Expert verified
The solutions are \(\sqrt{7}, -\sqrt{7}, 1, -1\).

Step by step solution

01

Understand the Absolute Value Equation

The equation to solve is \(\big|w^{2}-4\big|=3\). This means that \(w^{2}-4\) can be either 3 or -3 because the absolute value of a number is the distance from zero and is always positive.
02

Set Up the Two Possible Equations

Based on the definition of absolute value, we create two separate equations to solve: \(w^{2}-4=3\) and \(w^{2}-4=-3\).
03

Solve the First Equation \(w^{2}-4=3\)

Add 4 to both sides: \(w^{2}=7\). Then, take the square root of both sides: \(w = \pm \sqrt{7}\). The solutions to this equation are \(w = \sqrt{7}\) and \(w = -\sqrt{7}\).
04

Solve the Second Equation \(w^{2}-4=-3\)

Add 4 to both sides: \(w^{2}=1\). Then, take the square root of both sides: \(w = \pm 1\). The solutions to this equation are \(w = 1\) and \(w = -1\).
05

Combine All Solutions

Collect all possible solutions from both equations: \(w = \sqrt{7},\ -\sqrt{7}, 1, \ -1\).

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Key Concepts

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

solving absolute value equations
Solving absolute value equations involves finding unknowns within an absolute value expression. Absolute values denote the distance from zero, which is always positive. In the given problem \(\big|w^2 - 4\big| = 3\), you need to consider two scenarios because the expression inside the absolute value \((w^2 - 4)\) could equal 3 or -3. This leads to two equations: \w^2 - 4 = 3\ and \w^2 - 4 = -3\.
quadratic equations
Quadratic equations take the form \(ax^2 + bx + c = 0\), where \a, b,\ and \c\ are constants. The equations \w^2 - 4 = 3\ and \w^2 - 4 = -3\ fit this form. To solve them, you first isolate the quadratic term and then take the square root on both sides. Remember also that solving a quadratic in the form \w^2 = k\ yields two answers: \w = \sqrt{k}\ and \w = -\sqrt{k}.
roots of equations
The roots of an equation are the solutions that satisfy the equation. For a quadratic equation \(w^2 - 4 = 3\), you add 4 to both sides and get \(w^2 = 7\). By taking the square root of both sides, you find the roots: \w = \pm\sqrt{7}\. Similarly, solving \(w^2 - 4 = -3\) involves adding 4 to both sides to get \(w^2 = 1\) and taking the square root, yielding roots: \w = \pm1\.
square roots
Square roots are values that, when multiplied by themselves, give the original number. If you have \(w^2 = 7\), taking the square root results in \w = \pm\sqrt{7}\. Note the \pm\ symbol, indicating both positive and negative roots. So for \(w = 1\) or \(w = -1\), these are straightforward square roots since \(1^2 = 1\) and \((-1)^2 = 1\). Square roots are crucial for solving quadratic equations.

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