Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 9: Problem 77
Solve by completing the square. $$x^{2}-20 x=21$$
Short Answer
Expert verified
The solutions are \(x = 21\) and \(x = -1\).
Step by step solution
01
Rewrite the equation
Start by moving the constant term to the other side of the equation: \(x^2 - 20x - 21 = 0\) becomes \(x^2 - 20x = 21\)
02
Identify the coefficient of the linear term
Identify the coefficient of the linear term, which is the term with \(x\): Here, the coefficient is -20.
03
Compute half of the coefficient and square it
Take half of the coefficient of \(x\), then square it: \text{Half of -20} = -10 \text{Square of -10} = 100
04
Add and subtract the square inside the equation
Add and subtract 100 inside the equation to complete the square: \(x^2 - 20x + 100 - 100 = 21\) This simplifies to: \((x^2 - 20x + 100) - 100 = 21\)
05
Rewrite the left side as a perfect square
Express the left side as a square of a binomial: \((x - 10)^2 - 100 = 21\)
06
Isolate the squared term
Solve for the squared term by moving -100 to the right side: \((x - 10)^2 = 21 + 100\) Simplifying this gives: \((x - 10)^2 = 121\)
07
Take the square root of both sides
Take the square root of both sides of the equation: \(x - 10 = \text{卤} \text{鈭�121}\)
08
Solve for x
Solve for \(x\) by isolating it: \(x - 10 = \text{卤} 11\) \(x = 10 + 11\) or \(x = 10 - 11\) \(x = 21\) or \(x = -1\)
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 equations
Quadratic equations are mathematical expressions of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. In the given exercise, we have the equation \(x^2 - 20x = 21\). Quadratic equations can be solved using various methods, such as:
- Factoring
- Using the quadratic formula \( \frac{-b \text{卤} \text{鈭�(b^2-4ac)}}{2a} \)
- Completing the square
perfect square trinomial
A perfect square trinomial is a quadratic expression that can be written as the square of a binomial. For example, \((x - 10)^2\) represents a perfect square trinomial. In the process of completing the square, we add and subtract a term to transform the quadratic equation into a perfect square trinomial. Here鈥檚 how:
- Start with the equation: \(x^2 - 20x = 21\)
- Identify the coefficient of \(x\), which is -20.
- Take half of this coefficient: \(\frac{-20}{2} = -10\)
- Square this result: \((-10)^2 = 100\)
- Add and subtract this square inside the equation: \(x^2 - 20x + 100 - 100 = 21\).
- Rewrite it as: \((x - 10)^2 - 100 = 21\).
solving equations
Once we have completed the square and obtained the equation \((x - 10)^2 = 121\), the next step is to solve for \(x\). Here's the step-by-step process:
- Isolate the squared term: \((x - 10)^2 = 121\)
- Take the square root of both sides: \(x - 10 = \text{卤} \text{鈭�121}\)
- This gives us two equations: \(x - 10 = 11\) and \(x - 10 = -11\)
- Solve for \(x\): For the first equation: \(x = 21\); for the second equation: \(x = -1\)
