/*! 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 1 Solve each equation. $$a^{2}=4... [FREE SOLUTION] | 91影视

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Chapter 9: Problem 1

Solve each equation. $$a^{2}=49$$

Short Answer

Expert verified
a = 7 or a = -7

Step by step solution

01

Identify the Equation

The given equation is \(a^{2} = 49\). To find the value of a, isolate the variable.
02

Take the Square Root

To isolate \(a\), take the square root of both sides of the equation. This gives: \[\sqrt{a^{2}} = \sqrt{49}\] When taking the square root of both sides, remember to consider both the positive and negative roots.
03

Consider Both Roots

Taking the square root of both sides yields: \[a = \pm 7\] This means that \(a\) can be either \(7\) or \(-7\).

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Key Concepts

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

Square Roots
The concept of square roots is essential in solving quadratic equations. A square root of a number is a value that, when multiplied by itself, gives the original number. For example, both 7 and -7 are square roots of 49 because \(7 \times 7 = 49\) and \(-7 \times -7 = 49\). In mathematical terms, the square root of a number x is written as \sqrt{x}\. When solving equations like \(a^2 = 49\), taking the square root of both sides helps to isolate the variable, which is a key step in finding the solutions. Always remember to consider both the positive and negative roots when solving equations involving square roots. This ensures you capture both possible solutions.
Positive and Negative Roots
When you take the square root of a number in a quadratic equation, it's crucial to remember that there are always two roots: positive and negative. For example, in the equation \(a^2=49\), taking the square root of both sides gives \sqrt{a^2} = \sqrt{49}\. This yields \a = \pm 7\, which means \a\ can be either 7 or -7. Ignoring one of these roots would miss out on a possible solution. Understanding this concept helps you to accurately solve quadratic equations and ensures you find all potential solutions to the problem.
Isolating Variables
Isolating the variable is a crucial step in solving equations. The goal is to have the variable by itself on one side of the equation. In the equation \(a^2 = 49\), the variable \a\ is squared. To isolate \a\, you need to get rid of the square by taking the square root of both sides, resulting in \sqrt{a^2} = \sqrt{49}\. This simplifies to \a = \pm 7\. Isolating the variable helps simplify the equation and makes it easier to solve. It's essential to use operations like taking square roots, adding, subtracting, multiplying, or dividing to achieve this isolation effectively.

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Most popular questions from this chapter

Rewrite each function in the \(f(x)=a(x-h)^{2}+k\) form by completing the square. $$f(x)=3 x^{2}+6 x-1$$

(a) rewrite each function in \(f(x)=a(x-h)^{2}+k\) form and (b) graph it by using transformations. $$f(x)=-x^{2}+8 x-16$$

Graph each function using a horizontal shift. $$f(x)=(x-2)^{2}$$

Solve. Round answers to the nearest tenth. A family of three young children just moved into a house with a yard that is not fenced in. The previous owner gave them 300 feet of fencing to use to enclose part of their backyard. Use the quadratic function \(A(x)=x\left(150-\frac{x}{2}\right)\) determine the maximum area of the fenced in yard.

Graph each function using a horizontal shift. $$f(x)=(x+3)^{2}$$
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