/*! 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 41 \(y^{2}-18=7\)... [FREE SOLUTION] | 91影视

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Chapter 9: Problem 41

\(y^{2}-18=7\)

Short Answer

Expert verified
y = \pm 5

Step by step solution

01

- Isolate the quadratic term

To isolate the quadratic term, add 18 to both sides of the equation: \(y^{2} - 18 + 18 = 7 + 18\)
02

- Simplify the equation

Simplify the equation by combining like terms: \(y^{2} = 25\)
03

- Solve for y

To solve for \(y\), take the square root of both sides of the equation: \(y = \pm \sqrt{25}\)
04

- Simplify the square root

Simplify the square root: \(y = \pm 5\)

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Key Concepts

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

Isolating the quadratic term
In this problem, we start with the equation \(y^2 - 18 = 7\).
The goal is to isolate the quadratic term, which is \(y^2\), on one side of the equation.
To do this, we need to move all other terms to the opposite side.
We can achieve this by adding 18 to both sides of the equation.
This process involves performing the same operation on each side to maintain the equality.
So, we get: \(y^2 - 18 + 18 = 7 + 18\) which simplifies to \(y^2 = 25\).
This step is crucial because it sets up a simpler equation where the variable term stands alone.
Taking the square root
Now that we have isolated the quadratic term, the next step is to solve for \(y\).
To do this, we need to take the square root of both sides of the equation \(y^2 = 25\).
Taking the square root will help us find the value of \(y\) that satisfies the equation.
The square root operation is indicated by the symbol \( \sqrt{ } \), and it can yield both positive and negative results.
Therefore, when we take the square root of both sides, we get: \(y = \pm \sqrt{25}\).
This means \(y\) can be both positive and negative values that, when squared, will be equal to 25.
Combining like terms
Combining like terms is a primary algebraic skill.
In this problem, after isolating the quadratic term, we combined the constant terms on the right side.
Here鈥檚 how: when we have \(7 + 18\), we simply add the numbers together to simplify the equation.
This gives us \(25\).
In combining like terms, you focus on adding or subtracting terms with the same variable and degree.
In this case, there were only constants to combine, simplifying our equation to \(y^2 = 25\).
This combined form makes it easier to proceed to the next steps efficiently.

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Most popular questions from this chapter

\(p^{2}+7=-57\)

\(m^{2}+24 m+150=0\)

\(5 d^{2}-3 d+2=0\)

Problem: Find the \(x\)-intercepts of the graph of \(y=x^{2}+2 x-24\) Incorrect Answer: \(0=(x-6)(x+4)\) $$ \begin{array}{rrr} x-6=0 & \text { or } & x+4=0 \\ +6+6 & & -4=-4 \\ \hline x+0=6 & x+0=-4 \\ x=6 & x=-4 \end{array} $$ The \(x\)-intercepts are \((6,0)\) and \((-4,0)\).

\(c^{2}+8=-15\)
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