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Chapter 9: Problem 69

In your own words, explain the Square Root Property.

Short Answer

Expert verified
The Square Root Property solves quadratic equations by taking the square root of both sides, yielding two solutions: one positive and one negative.

Step by step solution

01

Understanding the Square Root Property

The Square Root Property is a method used to solve quadratic equations of the form \( ax^2 = c \) by taking the square root of both sides. This can be expressed as \( x^2 = c \) implying \( x = \pm \sqrt{c} \).
02

Applying the Square Root Property

To use the Square Root Property to solve an equation, isolate the term with the variable squared by dividing both sides of the equation by the coefficient of the squared term if necessary.
03

Taking the Square Root of Both Sides

Once isolated, take the square root of both sides of the equation. This results in two possible solutions, \( x = \sqrt{c} \) and \( x = -\sqrt{c} \).
04

Considering the Sign

Remember to include both the positive and negative square roots when solving for the variable.
05

Example

For example, if we have \( x^2 = 9 \), then applying the Square Root Property gives us two solutions: \( x = \sqrt{9} \) which is 3, and \( x = -\sqrt{9} \) which is -3. Hence, the solution is \( x = \pm 3 \).

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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 \[ ax^2 + bx + c = 0 \]. These equations represent parabolas when graphed. Solving quadratic equations can be tricky, but certain methods, like the Square Root Property, make it easier. The Square Root Property is especially useful for simple quadratics of the form \[ ax^2 = c \].
To apply the Square Root Property:
  • First, ensure the equation is in the form \( x^2 = c \)
  • Then, take the square root of both sides, giving \( x = \pm \sqrt{c} \)
By following these steps, you can solve many quadratic equations efficiently. This method simplifies the process, allowing you to find solutions quickly.
Isolating Variable Terms
To solve a quadratic equation using the Square Root Property, you first need to isolate the term involving the variable squared. If the equation is in a more generalized form, like \[ ax^2 + bx + c = 0 \], you may need to simplify it first.
Here鈥檚 how to isolate the variable term:
  • Move all terms except the squared term to the other side of the equation.
  • Divide by any coefficient that multiplies the squared term, to get the equation in the form \( x^2 = c \).
For example, if you have \[ 4x^2 = 36 \], divide both sides by 4 to get \[ x^2 = 9 \]. Isolating the term simplifies the equation and prepares it for using the Square Root Property.
Positive and Negative Roots
When solving quadratic equations with the Square Root Property, remember that taking the square root of both sides results in two solutions鈥攐ne positive and one negative.
Whenever you solve equations of the form \[ x^2 = c \], you get \[ x = \sqrt{c} \] and \[ x = -\sqrt{c} \].
This is because both \( \sqrt{c} \) and \( -\sqrt{c} \) squared will give you \( c \) back.
For instance, for \[ x^2 = 9 \], both 3 and -3 are valid solutions because \[ 3^2 = 9 \] and \[ (-3)^2 = 9 \].
  • Always consider both the positive and negative roots when solving quadratic equations.
  • This ensures you find all possible solutions to the equation.
Ignoring one of the roots means you might miss a crucial part of the solution, so always check both roots.

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Most popular questions from this chapter

Find the maximum or minimum value of each function. $$y=4 x^{2}-49$$

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