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Chapter 9: Problem 9

\(w^{2}=484\)

Short Answer

Expert verified
w = 22 or w = -22.

Step by step solution

01

Identify the mathematical equation

The given equation is a quadratic equation: \(w^{2}=484\).
02

Isolate the squared variable

We already have the squared variable isolated: \(w^{2} = 484 \).
03

Take the square root of both sides

To solve for \(w\), take the square root of both sides of the equation: \[ \sqrt{w^{2}} = \sqrt{484} \].
04

Simplify the square roots

Simplify both sides: \[ w = \pm 22 \].

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Key Concepts

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

Quadratic Equation
A quadratic equation is an equation of the form ax^2 + bx + c = 0 where *a*, *b*, and *c* are constants, and *x* represents an unknown variable. The term *a* ( the coefficient of x^2) must be non-zero. In this specific exercise, the given equation w^2 = 484 is a simple quadratic equation where a = 1, b = 0, and c = -484.
To solve a quadratic equation means to find the values of *x* (or *w* in our case) that make the equation true.
Quadratic equations often have two solutions because of the x^2 term.
In our example, we're solving w^2 = 484, which is brought into the context through the lens of a quadratic equation.
Square Root
The square root is a value that, when multiplied by itself, gives the original number. The square root of 484 is a number that, when squared (multiplied by itself), equals 484.
The symbol for the square root is \[ \sqrt{} \], so the square root of 484 is written as \[ \sqrt{484} \] .
Square roots can have both positive and negative values because (-a) \times (-a) = a^2. In our exercise, solving \[\sqrt{w^2} = \sqrt{484} \] results in w = \pm 22. Here, \pm indicates both +22 and -22 as potential solutions. It鈥檚 important to always consider both the positive and negative square roots while solving such equations.
Isolating Variable
Isolating the variable means to get the unknown variable (in this case, *w*) alone on one side of the equation. This step simplifies the equation and helps in systematically solving for the variable.
In the given exercise, the variable w^{2} is already isolated on one side of the equation w^{2} = 484.
Having the variable isolated simplifies the task to the next step, which is taking the square root of both sides. Isolating the variable typically involves:
  • Adding, subtracting, multiplying, or dividing both sides of the equation by the same number.
  • Applying algebraic identities or inverse operations.
Isolating the variable helps break down complex equations into manageable steps, where we can now apply mathematical operations (like square roots) to both sides.

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Most popular questions from this chapter

\(3 x^{2}-12 x=0\)

\(7 a^{2}=-252\)

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\(\sqrt{8^{2}-4(4)(8)}\)

\(v^{2}=-50\)
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