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Chapter 11: Problem 36

Solve using the square root property. Simplify all radicals. $$ x^{2}-13=0 $$

Short Answer

Expert verified
The solutions are \( x = \pm\sqrt{13} \).

Step by step solution

01

Isolate the Variable Term

Start by isolating the term that contains the variable. Add 13 to both sides of the equation:\[ x^{2} - 13 + 13 = 0 + 13 \] which simplifies to:\[ x^{2} = 13 \]
02

Apply the Square Root Property

Take the square root of both sides of the equation to solve for \( x \). Remember to include both the positive and negative roots:\[ \sqrt{x^{2}} = \pm\sqrt{13} \] which simplifies to: \[ x = \pm\sqrt{13} \]
03

Simplify the Radicals (if necessary)

In this case, \( \sqrt{13} \) is already in its simplest form because 13 is a prime number.

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Key Concepts

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

Square Root Property
When solving quadratic equations, the square root property is a powerful tool. It allows us to directly solve for the variable by taking the square root of both sides of the equation. For example, if we have \( x^2 = a \), we can solve it by taking the square root of both sides, giving us \( x = \pm \sqrt{a} \). Remember, you always need to consider both the positive and negative roots when applying this property because squaring either a positive or negative number will result in the same value.
Isolating Variable
To effectively apply the square root property, we must first isolate the variable term. This step involves rearranging the equation so that the term containing the variable is by itself on one side of the equation. In the exercise \( x^2 - 13 = 0 \), we added 13 to both sides to isolate the \( x^2 \) term, resulting in the equation \( x^2 = 13 \). This ensures the equation is ready for the next steps.
  • Start by adding, subtracting, multiplying, or dividing to isolate the variable term.
  • Ensure the variable term has a positive coefficient when isolated.
  • Make sure no other terms or constants are on the same side as the variable term.
Taking these steps correctly is crucial for accurately applying mathematical properties like the square root property.
Simplifying Radicals
Simplifying radicals means expressing the root in its simplest form. Radicals should have the smallest possible number inside the radical sign. For our problem \( x = \pm \sqrt{13} \), \( \sqrt{13} \) is already simplified because 13 is a prime number and cannot be broken down further. To simplify other radicals:
  • Factor the number under the radical sign into its prime factors.
  • Group the factors to pull out squares.
  • Simplify the expression by taking the square root of the groups and pulling them out of the radical.
For example, if you had \( \sqrt{20} \), you would break it down to \( \sqrt{4 \cdot 5} \) and simplify to \( 2 \sqrt{5} \). This process ensures the expression is manageable and easier to work with.

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Most popular questions from this chapter

Solve using the square root property. Simplify all radicals. $$ (4 x-3)^{2}=9 $$

Solve each equation. (All solutions are nonreal complex numbers.) $$ (6 k-1)^{2}=-8 $$

Use the quadratic formula to solve each equation. (All solutions for these equations are non real complex numbers.) $$ x(3 x+4)=-2 $$

Solve using the square root property. Simplify all radicals. $$ (3 x+2)^{2}=49 $$

William Froude was a 19th century naval architect who used the following expression, known as the Froude number, in shipbuilding. $$ \frac{v^{2}}{g \ell} $$ This expression was also used by R. McNeill Alexander in his research on dinosaurs. (Data from "How Dinosaurs Ran," Scientific American.) Use this expression to find the value of \(v\) (in meters per second), given \(g=9.8 \mathrm{~m}\) per sec \(^{2}\) (Round to the nearest tenth.) Triceratops: \(\ell=2.8\) Froude number \(=0.16\)
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