/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 89 Solve the equation. $$(t-19)^{... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 89

Solve the equation. $$(t-19)^{2}=0$$

Short Answer

Expert verified
The solution to the equation \((t-19)^{2}=0\) is \(t=19\).

Step by step solution

01

Analyzing the Quadratic equation

Analyze the given quadratic equation \((t-19)^{2}=0\). This is already a squared term equal to zero which implies that the term inside the bracket, i.e., \(t-19\) is equal to zero.
02

Solving the equation

Solve the equation by setting \(t-19=0\). To solve this, add 19 to both sides of the equation to isolate \(t\). Doing this results to \(t=19\).
03

Checking the solution

Check your answer by substituting \(t=19\) into the original equation. The equation becomes \((19-19)^{2}=0\), that reduces to \(0^{2}=0\). Since both sides match, our solution is correct.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Quadratic Equation
Quadratic equations are a type of polynomial equation that take the general form of
\[ ax^2 + bx + c = 0 \]
with \( a \), \( b \), and \( c \) being constants, and \( a \) not equal to zero since that would render the equation linear rather than quadratic. In our exercise, the given equation takes the shape of
\[ (t-19)^2 = 0 \]
which is a quadratic equation with imaginary values of \( a = 1 \), \( b = -38 \), and \( c = 361 \). However, it's been simplified to a perfect square set equal to zero. This compact form is a clue that our solution will be straightforward, leading to a single value for \( t \) that makes the equation true.
Algebraic Operations
Algebraic operations include processes like addition, subtraction, multiplication, division, and the rules of exponentiation used to manipulate algebraic expressions and solve equations. When solving for \( t \) in the given quadratic equation, we're demonstrating a basic but crucial algebraic operation: isolating the variable. By adding 19 to both sides of the equation \( t-19 = 0 \), we use the additive property of equality to keep the equation balanced while moving the constant term to the other side, simplifying to \( t = 19 \).
These operations are the fundamental tools we use to solve equations and better understand the relationships between numbers and symbols in algebra.
Solution Verification
Solution verification is vital in mathematics to confirm that our proposed answer indeed satisfies the original equation. By substituting the value of \( t = 19 \) back into the equation and simplifying, we should arrive at a true statement if the solution is correct. If we plug \( t = 19 \) into our quadratic equation, we get
\[ (19-19)^2 = 0 \]
which simplifies to \( 0^2 = 0 \), a true statement. This means \( t = 19 \) is the correct solution to the equation. Verification ensures that any errors made in the algebraic process are caught and that our answer is mathematically sound.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Solve the equation. $$(4 n-6)^{3}=0$$

Sketch the graph of the inequality. \(x+y<9\)

In Exercises \(69-72,\) you are tutoring a friend and want to create some quadratic equations that can be solved by factoring. Find a quadratic equation that has the given solutions and explain the procedure you used to obtain the equation. $$8 and - 8$$

Which of the following is a correct factorization of \(72 x^{2}-24 x+2 ?\) (A) \(-9(3 x-1)^{2}\) (B) \(8\left(9 x-\frac{1}{2}\right)^{2}\) (C) \(8\left(3 x-\frac{1}{2}\right)\left(3 x-\frac{1}{2}\right)\) (D) \(-8\left(3 x-\frac{1}{2}\right)^{2}\)

Which one of the following equations cannot be solved by factoring with integer coefficients? (A) \(12 x^{2}-15 x-63=0\) (B) \(12 x^{2}+46 x-8=0\) (C) \(6 x^{2}-38 x-28=0\) (D) \(8 x^{2}-49 x-68=0\)
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.