Problem 15 - Algebra for College Students

Question

Use the even-root property to solve each equation. $$x^{2}=81$$

Step-by-Step Solution

Answer

The solutions are x = 9 and x = -9.

Step 1: Understand the Even-Root Property

The even-root property states that if a^2 = b , then a = sqrt{b} or a = - sqrt{b} . This means you can take the square root of both sides of the equation to solve for the variable.

Step 2: Apply the Even-Root Property

Given the equation x^{2} = 81 , take the square root of both sides. Remember to consider both the positive and negative roots.

Step 3: Solve for x

Taking the square root of both sides gives: x = sqrt{81}orx = - sqrt{81} . Calculating the square root of 81, we get: x = 9 or x = -9

Key Concepts

Understanding the Even-Root Property

The even-root property is a useful tool to solve quadratic equations. If you have an equation of the form \(a^2 = b\), you can conclude that \(a = \sqrt{b}\) or \(a = -\sqrt{b}\). In simpler terms, by taking the square root of both sides of the equation, you can solve for the variable. This property is especially handy when the quadratic equation can be reduced to a perfect square.

How to Work with Square Roots

Square roots are found when you want to determine what number, when multiplied by itself, gives you the original number. For example, \(\sqrt{81} = 9\) because \(9 \times 9 = 81\). To solve for a variable, you need to take the square root of both sides of the equation. Always remember to consider two potential solutions: one positive and one negative. In the example provided:xc considerable is that same rules apply for all squares.
\(x^2 = 81\) becomes \(x = \sqrt{81}\) or \(x = -\sqrt{81}\).

• Calculating the square root of 81, get:
  \(x = 9\) or \(x = -9\).

Positive and Negative Roots

When you solve quadratic equations using the even-root property, always consider both positive and negative solutions. This is because every positive real number has two square roots: one positive and one negative. For example, \(\sqrt{81} = 9\) and \(\sqrt{81} = -9\). This duality comes from the fact that multiplying two negative numbers results in a positive number, hence both \(9 \times 9\) and \((-9) \times (-9)\) equal 81.
To summarize the steps in using the even-root property and working with square roots, remember:

• Take the square root of both sides
• Consider both the positive and negative roots
• Solve for the variable

By keeping these points in mind, solving quadratic equations can be simpler and more intuitive.