/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 15 \(-x^{2}=-169\)... [FREE SOLUTION] | 91影视

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Chapter 9: Problem 15

\(-x^{2}=-169\)

Short Answer

Expert verified
x = \pm 13

Step by step solution

01

Isolate the quadratic term

To begin solving the equation \(-x^{2} = -169\), divide both sides by \(-1\) to isolate the quadratic term. This will give you \(x^2 = 169\).
02

Apply the square root

Next, apply the square root to both sides of the equation \(x^2 = 169\). This results in \(x = \pm \sqrt{169}\).
03

Simplify the square root

Simplify \(\sqrt{169}\) to get \(\sqrt{169} = 13\). Therefore, \(x = \pm13\).

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Key Concepts

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

Isolating Variables
Isolating variables is the first step in solving equations, especially quadratic ones. It means getting the variable you are solving for, in this case, \(x\), on one side of the equation by itself. In the problem \-x^{2} = -169\, we start by removing the negative sign in front of the quadratic term. We do this by dividing both sides of the equation by -1. The equation then becomes \(x^2 = 169\). Now, we have successfully isolated the quadratic term which is necessary for the subsequent steps.
Square Root Method
Once the quadratic term is isolated, the next step is to eliminate the exponent. This is done using the square root method. When you have an equation like \(x^2 = 169\), you can find \(x\) by taking the square root of both sides of the equation. Applying the square root to both sides, we get: \(\[\begin{equation}\begin{aligned} x = \pm \sqrt{169} \end{aligned}\end{equation}\]\). The square root method is useful because it simplifies the equation to a linear form where \(x\) is by itself.
Simplifying Square Roots
Simplifying square roots is an essential step to get the final solution. After applying the square root to both sides, we need to simplify \(\[\begin{equation}\begin{aligned} x = \pm \sqrt{169} \end{aligned}\end{equation}\]\). Calculating the square root of 169, we find that \(\[\begin{equation}\begin{aligned} \sqrt{169} = 13 \end{aligned}\end{equation}\]\). So, the equation simplifies to \(x = \pm 13\). It's critical to understand how to simplify square roots accurately to solve equations efficiently.
Positive and Negative Solutions
When dealing with square roots in equations, it's important to remember they produce both positive and negative solutions. This is because both \(13 \mathbf{and} -13\) when squared will return 169. Therefore, our final solution is \(x = \pm 13\). This notation shows both possible values: \(x = 13\) \mathbf{and} \(x = -13\). Including both solutions ensures that we account for all possible cases in quadratic equations.

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