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Chapter 8: Problem 22

Find all complex-number solutions. $$ (x+1)^{2}=100 $$

Short Answer

Expert verified
The solutions are x = 9 and x = -11.

Step by step solution

01

Isolate the quadratic expression

Start by noting the given equation: o (x + 1)^2 = 100
02

Apply the square root on both sides

To solve for x + 1 , take the square root of both sides of the equation. Remember that you will get two solutions: one positive and one that is negative due to the square root property. So you have: sqrt((x + 1)^2) = sqrt(100)
03

Simplify the square root

Taking the square root of both sides, we get: x + 1 = 卤 鈭�100 Since 鈭�100 = 10 , we can write: x + 1 = 卤 10
04

Solve for x

Now, we solve for x by isolating it in both resultant equations: x + 1 = 10 x + 1 = -10 For the first equation: x + 1 = 10 x = 10 - 1 x = 9 For the second equation: x + 1 = -10 x = -10 - 1 x = -11

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Key Concepts

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

Quadratic Equations
Quadratic equations are polynomial equations of degree 2. They are written in the form:
\[ ax^2 + bx + c = 0 \].
In this equation, \( a \), \( b \), and \( c \) are constants, with \( a \) not equal to zero. This type of equation has a graph in the shape of a parabola.
Quadratic equations often have two solutions.
They can be solved using different methods:
  • Factoring
  • Completing the square
  • Quadratic formula
  • Graphing
Understanding these methods is essential to solve various algebra problems in mathematics.
Square Root Property
The square root property is a useful tool when dealing with quadratic equations.
It states that if \( x^2 = k \), then:
  • \( x = \pm \sqrt{k} \)
This property allows you to solve equations by taking the square root on both sides.
Remember, always include both the positive and negative roots. Here's how it works in practice using the equation:
\[ (x+1)^{2}=100 \]
  • Apply the square root to both sides: \( \sqrt{(x+1)^2} = \sqrt{100} \)
  • This yields: \( x+1 = \pm \sqrt{100} \)
  • Since \( \sqrt{100} = 10 \), we have: \( x+1 = \pm 10 \)
Using the square root property simplifies solving quadratics, making it a cornerstone of algebra techniques.
Solving Equations
Solving equations involves finding the value(s) of the variable(s) that satisfy the equation.
To solve a quadratic equation like \( (x+1)^{2}=100 \), follow these steps:
  • Isolate the quadratic expression.

  • Apply the square root to both sides, considering both the positive and negative roots: \( x+1 = \pm 10 \).

  • Solve for \( x \) by isolating it in each of the resulting equations:
  • For \( x+1 = 10 \) :
    \( x = 10 - 1 = 9 \).

  • For \( x+1 = -10 \) :
    \( x = -10 - 1 = -11 \).

Thus, the solutions to the equation \( (x+1)^{2}=100 \) are \( x = 9 \) and \( x = -11 \).
Practicing these steps improves your ability to solve various types of algebraic equations.

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Most popular questions from this chapter

A barge and a fishing boat leave a dock at the same time, traveling at a right angle to each other. The barge travels \(7 \mathrm{km} / \mathrm{h}\) slower than the fishing boat. After \(4 \mathrm{hr},\) the boats are \(68 \mathrm{km}\) apart. Find the speed of each boat.

It takes Deanna twice as long to set up a fundraising auction as it takes Donna. Together they can set up for the auction in 4 hr. How long would it take each of them to do the job alone?

Number of Diagonals. A polygon with \(n\) sides has \(D\) diagonals, where \(D\) is given by the function $$D(n)=\frac{n(n-3)}{2}$$ Find the number of sides \(n\) if $$27 \leq D \leq 230.$$

Solve each formula for the indicated letter. Assume that all variables represent positive numbers. \(a t^{2}+b t+c=0,\) for \(t\) (An algebraic formula)

Find the domain of each function. $$f(x)=\sqrt{9-x^{2}}$$
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