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Chapter 10: Problem 4

Solve the following quadratic equations. \(t^{2}-75=0\)

Short Answer

Expert verified
The solutions are \(t = 5\sqrt{3}\) and \(t = -5\sqrt{3}\).

Step by step solution

01

- Identify the equation

The given quadratic equation is: \[t^{2} - 75 = 0\]
02

- Isolate the quadratic term

Add 75 to both sides of the equation to isolate the quadratic term:\[t^{2} - 75 + 75 = 0 + 75\]This simplifies to:\[t^{2} = 75\]
03

- Solve for the variable

Take the square root of both sides to solve for \(t\):\[t = \pm \sqrt{75}\]Simplify the square root:\[t = \pm 5\sqrt{3}\]

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Key Concepts

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

Isolating the Variable
To solve a quadratic equation, we first need to isolate the variable. Let's break this down.

In equations like \(t^{2} - 75 = 0\), the term with the variable needs to stand alone on one side of the equation. In our example, this term is \(t^{2}\).

How do we do this? We add 75 to both sides of the equation.
This keeps the equation balanced, meaning both sides stay equal. So, \(t^{2} - 75 + 75 = 0 + 75\). This simplifies to \(t^{2} = 75\).
  • Always perform the same operation on both sides.
  • Check each step for accuracy.
Now, our equation has \(t^{2}\) isolated, and we are ready to move to the next step, which is solving for the variable.
Taking the Square Root
The next step in solving the quadratic equation is taking the square root of both sides.
This helps us find the value of the variable, t.

In our isolated equation \(t^{2} = 75\), we take the square root to solve for t:

\(t = \pm \sqrt{75}\). Remember, the \( \pm \) symbol indicates there are two possible solutions: positive and negative.
  • Use the square root symbol correctly.
  • Include both positive and negative solutions.
Taking the square root undoes the squaring process, making \(t = \pm \sqrt{75}\).
Simplifying Square Roots
To complete the solution, we simplify the square root. Simplifying \(\sqrt{75}\) means finding perfect squares within the number.

First, factorize 75 as \(25 \times 3\). We know that \(\sqrt{25} = 5\), so: \(\sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3}\).
  • Identify any perfect squares.
  • Extract the square root of those perfect squares.
Hence, \(t = \pm 5\sqrt{3}\). By simplifying the square root, we find the cleanest form of our solution.
Now you know how to solve and simplify a quadratic equation!

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Most popular questions from this chapter

In the following exercises, identify the most appropriate method (Factoring, Square Root, or Quadratic Formula) to use to solve each quadratic equation. Do not solve. (a) \(6 a^{2}+14=20\) (b) \(\left(x-\frac{1}{4}\right)^{2}=\frac{5}{16}\) (c) \(y^{2}-2 y=8\)

Make up a problem involving the product of two consecutive odd integers. Start by choosing two consecutive odd integers. (a) What are your integers? (b) What is the product of your integers? (c) Solve the equation \(n(n+2)=p,\) where \(p\) is the product you found in part (b). (1) Did you get the numbers you started with?

Solve by using the Quadratic Formula. \(8 n^{2}-3 n+3=0\)

In the following exercises, determine the number of solutions to each quadratic equation. a. \(25 p^{2}+10 p+1=0\) b.\(7 q^{2}-3 q-6=0\) c.\(7 y^{2}+2 y+8=0\) d.\(25 z^{2}-60 z+36=0\)

Solve by using the Quadratic Formula. \(\frac{1}{3} m^{2}+\frac{1}{12} m=\frac{1}{4}\)
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