Problem 52
Question
Solve using the Square Root Property. \((y-4)^{2}+10=9\)
Step-by-Step Solution
Verified Answer
No real solutions.
1Step 1 - Isolate the square term
Subtract 10 from both sides of the equation to isolate the squared term: (y-4)^2 + 10 - 10 = 9 - 10 This simplifies to: (y-4)^2 = -1
2Step 2 - Assess the possibility of solving
Notice that the equation (y-4)^2 = -1 contains a negative number on the right side. In the set of real numbers, no real number squared results in a negative value. Hence, the equation has no real solutions.
Key Concepts
Isolating the Square TermSolving EquationsReal Numbers
Isolating the Square Term
When solving an equation using the Square Root Property, the first step is to isolate the term that is squared. This means you want the term with the square to be alone on one side of the equation.
For example, in the equation \((y-4)^{2}+10=9\), we need to move the constant term to the other side of the equation.
To do this, we subtract 10 from both sides:
\((y-4)^2 + 10 - 10 = 9 - 10\)
This simplifies to: \((y-4)^2=-1\).
By isolating the square term, we set ourselves up for the next steps in solving the equation.
For example, in the equation \((y-4)^{2}+10=9\), we need to move the constant term to the other side of the equation.
To do this, we subtract 10 from both sides:
\((y-4)^2 + 10 - 10 = 9 - 10\)
This simplifies to: \((y-4)^2=-1\).
By isolating the square term, we set ourselves up for the next steps in solving the equation.
Solving Equations
After isolating the square term, the typical next step is to apply the Square Root Property. This property states that if \(x^2=a\), then \(( x = \sqrt{a}\text{ or } -\sqrt{a} )\).
However, it's important to check if it's possible to take the square root of the term on the other side of the equation.
In our example \((y-4)^2 = -1\), we have a negative number on the right side.
Real numbers do not include the square roots of negative numbers. Hence, solving such an equation is impossible within the real numbers.
However, it's important to check if it's possible to take the square root of the term on the other side of the equation.
In our example \((y-4)^2 = -1\), we have a negative number on the right side.
Real numbers do not include the square roots of negative numbers. Hence, solving such an equation is impossible within the real numbers.
Real Numbers
Real numbers are a set of numbers that include all the numbers on the number line.
This set includes integers, fractions, and irrational numbers.
However, real numbers exclude imaginary numbers, which are derived when taking the square root of a negative number.
In our exercise \((y-4)^2=-1\), the right side of the equation is negative.
No real number, when squared, will ever give a negative result. Therefore, \((y-4)^2=-1\) has no real solutions.
Understanding the limitations of real numbers is crucial for solving equations correctly.
This set includes integers, fractions, and irrational numbers.
However, real numbers exclude imaginary numbers, which are derived when taking the square root of a negative number.
In our exercise \((y-4)^2=-1\), the right side of the equation is negative.
No real number, when squared, will ever give a negative result. Therefore, \((y-4)^2=-1\) has no real solutions.
Understanding the limitations of real numbers is crucial for solving equations correctly.
Other exercises in this chapter
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