Problem 41 - Beginning and Intermediate Algebra

Question

Solve using the square root property. Simplify all radicals. $$ 2 t^{2}+7=61 $$

Step-by-Step Solution

Answer

t = Respective values based on sign change

Step 1: Isolate the quadratic term

First, subtract 7 from both sides of the equation to isolate the term with the variable.\[ 2t^2 + 7 - 7 = 61 - 7 \]This simplifies to:\[ 2t^2 = 54 \]

Step 2: Divide by the coefficient of the squared term

Next, divide both sides by 2 to solve for \(t^2\).\[ \frac{2t^2}{2} = \frac{54}{2} \]This simplifies to:\[ t^2 = 27 \]

Step 3: Apply the square root property

Take the square root of both sides to solve for \(t\).\[ t = \,\pm \, \frac{27}{t} \]Simplifying the right-hand side, you need to break down the square root into factors of perfect squares and simplify if possible:\[ t = 3\beta h^2 \]

Key Concepts

Square Root Property

When you see a quadratic equation in the form of \( ax^2 = c \), you can solve for \( x \) using what is known as the *square root property*. This property allows you to take the square root of both sides of the equation. However, remember that taking the square root introduces two potential solutions: one positive and one negative.
For instance, if we have \( t^2 = 27 \), we apply the square root property to get:
\[ t = \pm \sqrt{27} \]
This yields two solutions because both positive and negative values, when squared, will result in 27.

Simplifying Radicals

Radicals can often be simplified by identifying perfect square factors within the number under the radical symbol. When you have a number like \( \sqrt{27} \), you can break it down into its factors:.

• 27 is the same as 9 * 3
• Thus, we can write \( \sqrt{27} \) as \( \sqrt{9} \times \sqrt{3} \)

Since 9 is a perfect square, it simplifies to 3:
\[ \sqrt{27} = \sqrt{9 \times 3} = \sqrt{9} \times \sqrt{3} = 3\sqrt{3} \]
So, instead of \( \pm \sqrt{27} \), you get:\
\[ t = \pm 3\sqrt{3} \]

Isolating the Quadratic Term

The first key step in solving any quadratic equation is to isolate the term which includes the variable raised to the power two. For example, in the equation:
\[ 2t^2 + 7 = 61 \]
You would first subtract 7 from both sides to get:
\[ 2t^2 = 54 \]
Next, divide by the coefficient of the quadratic term (which is 2 in this case) to isolate \( t^2 \).
\[ t^2 = \frac{54}{2} = 27 \] This step is crucial because it simplifies the equation, making it possible to apply other properties like the *square root property* later.

Perfect Squares

A *perfect square* is a number that can be expressed as the product of an integer with itself. Recognizing perfect squares can simplify solving quadratic equations and working with radicals. Examples of perfect squares include:

• \( 1^2 = 1 \)
• \( 2^2 = 4 \)
• \( 3^2 = 9 \)
• \( 4^2 = 16 \)

In the context of simplifying radicals, knowing your perfect squares can help you break down the numbers. For instance, recognizing that 9 is a perfect square helped us simplify \( \sqrt{27} \) earlier as \( 3\sqrt{3} \).
This knowledge is particularly valuable when dealing with quadratic equations, as you often encounter such perfect squares during the simplification process.