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Chapter 1: Problem 93

Solve the equation by using any method. \((2 y+7)(y+1)=2 y^{2}-11\)

Short Answer

Expert verified
y = -2.

Step by step solution

01

Expand the left side of the equation

First, distribute the terms on the left-hand side. Start with o\((2 y+7)(y+1) $$= 2y \times y + 2y \times 1 + 7 \times y + 7 \times 1\)o \((2 y+7)(y+1) = 2y^2 + 2y + 7y + 7\) =\(2y^2 + 9y + 7\).
02

Set the equation to standard form

Now that the left-hand side is expanded, set the equation equal to 0 by subtracting all terms on the right-hand side from both sides of the equation. \ \ \((2y^2 + 9y + 7 - 2y^2 + 11 = 0\) ). Simplify: $$9y + 18 = 0$$.
03

Solve for \(y\)

Solve the linear equation. Move the constant term to the other side: $$9y = - 18$$.Divide both sides by 9: $$y = -2.$$.

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Key Concepts

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

Equation Expansion
When solving a quadratic equation like \( (2y+7)(y+1)=2y^2-11 \), the first step often involves expanding the equation. Equation expansion means distributing the terms inside the parentheses to simplify the expression.

In this example, expand \( (2 y+7)(y+1) \) by distributing each term in the first parenthesis by each term in the second parenthesis:
\[ (2y+7)(y+1) = 2y \times y + 2y \times 1 + 7 \times y + 7 \times 1 \]

This gives us:
\[ 2y^2 + 2y + 7y + 7 \]

Then combine like terms to simplify it further:
\[ 2y^2 + 9y + 7 \]

This step removes the parentheses and sets us up to work with each term more easily. Remember to multiply each term carefully and then add like terms.
Standard Form Conversion
After expanding the equation, the next step is converting the equation to standard form. The standard form of a quadratic equation is \( ax^2 + bx + c = 0 \).

From our expanded equation \( 2 y^2 + 9 y + 7 \), we need to subtract \( 2 y^2 - 11 \) on both sides to set it to zero:
\[ 2y^2 + 9y + 7 - 2y^2 + 11 = 0 \]

Simplify the expression by combining like terms:
\[ 9y + 18 = 0 \]

Now, we have transformed our equation into a linear equation, which is much simpler to solve. Converting to standard form is an essential step because it provides a structured way to isolate the variable.
Linear Equation Solving
Finally, solving the linear equation \( 9y + 18 = 0 \) involves isolating the variable \( y \).

Begin by moving the constant term to the other side of the equation:
\[ 9 y = -18 \]

Then, divide both sides by the coefficient of \( y \):
\[ y = -18 / 9 \]

Simplify the division:
\[ y = -2 \]

This straightforward method of isolating \( y \) and solving for it highlights the power of converting a quadratic equation into a more manageable linear equation. Always check your solution by plugging it back into the original equation to verify its correctness.

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Most popular questions from this chapter

A box of cereal is labeled to contain 16 oz. A consumer group takes a sample of 50 boxes and measures the contents of each box. The individual content of each box differs slightly from 16 oz, but by no more than 0.5 oz. a. If \(x\) represents the exact weight of the contents of a box of cereal, write an absolute value inequality that represents an interval in which to estimate \(x\). b. Solve the inequality and interpret the answer.

Determine the set of values for \(x\) for which the radical expression would produce a real number. For example, the expression \(\sqrt{x-1}\) is a real number if \(x-1 \geq 0\) or equivalently, \(x \geq 1\). a. \(\sqrt{x+4}\) b. \(\sqrt[3]{x+4}\)

Solve the equations. \(|3 z|=\left|\frac{1}{3} z\right|\)

Solve the inequality and write the solution set in interval notation. Solve the inequality for \(\mu:|\mu-\bar{x}|<\frac{z \sigma}{\sqrt{n}}\). (Do not rationalize the denominator.)

a. Write an absolute value equation or inequality to represent each statement. b. Solve the equation or inequality. Write the solution set to the inequalities in interval notation. The variation between the measured value \(t\) and 60 min is less than \(0.2 \mathrm{~min}\).
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