/*! 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 32 Solve each equation by the Squar... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 32

Solve each equation by the Square Root Method. $$ x^{2}=36 $$

Short Answer

Expert verified
The solutions are \( x = 6 \) and \( x = -6 \).

Step by step solution

01

Understand the Equation

The given equation is \( x^{2} = 36 \). This equation is a quadratic equation where we need to find the value of \( x \) such that when \( x \) is squared, it equals 36.
02

Take the Square Root of Both Sides

To solve for \( x \), take the square root of both sides of the equation. This gives us: \( \sqrt{x^{2}} = \sqrt{36} \).
03

Simplify the Equation

The square root of \( x^{2} \) is \( x \), and the square root of 36 is 6. Remember that taking the square root of a number can give both a positive and negative solution. Therefore, we have two potential solutions: \( x = 6 \) and \( x = -6 \).

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Key Concepts

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

solving quadratic equations
Quadratic equations are equations of the form \(ax^2 + bx + c = 0\). In our exercise, the equation \(x^2 = 36\) is a simple form of a quadratic equation where \( a = 1 \), \( b = 0 \), and \( c = -36 \). To solve a quadratic equation, we often need to find the values of \( x \) that satisfy this equation. For our specific problem, we can use the Square Root Method, a straightforward and efficient method for equations where \( b = 0 \). This method involves isolating \( x^2 \) on one side and then taking the square root of both sides.
taking square roots
Taking the square root is the process of finding a number which, when multiplied by itself, gives the original number. In our equation \( x^2 = 36 \), we take the square root of both sides to solve for \( x \). The square root of \( x^2 \) ( \( \sqrt{x^2} \)) is \( x \), because \( x \times x = x^2 \). The square root of 36 is 6, given that \( 6 \times 6 = 36 \). Hence, we write: \( \sqrt{x^2} = \sqrt{36} \). Simplifying this, we get \( x = 6 \). But there is another important aspect to consider: the solutions can also be negative.
positive and negative solutions
When taking the square root of a number, it's crucial to remember that there are always two potential solutions: one positive and one negative. This is because both \( 6 \times 6 = 36 \) and \( -6 \times -6 = 36 \). Therefore, from our equation: \( x = \pm 6 \), we get the two solutions \( x = 6 \) and \( x = -6 \). This dual nature of square roots ensures we account for all possible values of \( x \) that satisfy the original equation. Always remember to include both solutions when solving similar quadratic equations using the square root method.

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Most popular questions from this chapter

Find the real solutions, if any, of each equation. Use any method. $$ x^{2}-5=0 $$

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