Problem 94

Question

Solve each equation in Exercises $83-108$ by the method of your choice. $$ (2 x+7)^{2}=25 $$

Step-by-Step Solution

Verified

Answer

The solutions for the equation $(2 x+7)^{2}=25$ are $x = -1$ and $x = -6$.

1Step 1: Take the square roots

First, take the square root of both sides of the equation. This yields two equations: \[2x+7 = \sqrt{25}\] and \[2x+7 = -\sqrt{25}\].

2Step 2: Solve the square root equations

Now, these equations are solved: For \[2x+7 = \sqrt{25}\], this simplifies to \[2x+7 = 5\] and for \[2x+7 = -\sqrt{25}\], this simplifies to \[2x+7 = -5\].

3Step 3: Solve for x

Solving these equations for $x$ now: For the first equation \[2x+7 = 5\], this simplifies to \[2x = -2\] or \[x = -1\]; and for the second equation \[2x+7 = -5\], this simplifies to \[2x = -12\] or \[x = -6\].

Key Concepts

Square Root MethodSolving EquationsAlgebraic Manipulation

Square Root Method

The square root method is a helpful strategy to solve certain types of quadratic equations. Especially those that can be expressed in the form $(ax + b)^2 = c$. The method starts by taking the square root of both sides of the equation. This effectively "undoes" the square since squaring and square-rooting are inverse operations. When you apply the square root on both sides, remember that there are always two possible roots: the positive root and the negative root.
In practice, if you are working with an equation like $(2x+7)^2=25$, you take the square roots to get $2x+7 = \pm 5$. The "$\pm$" symbol means two separate equations:

$2x+7 = 5$
$2x+7 = -5$

These represent the two possible values for $2x+7$ that satisfy the original quadratic equation.

Solving Equations

Solving equations involves finding the values of the variables that make the equation true. When you've taken the square root in a quadratic equation, you end up with simpler linear equations.
For your example: After you take the square roots of $(2x+7)^2=25$, you obtain:

$2x+7 = 5$
$2x+7 = -5$

These are simpler because they are linear equations. Linear equations can be solved by isolating the variable. This means performing operations on both sides to leave the variable on one side of the equation. Keep in mind that operations should maintain the balance of the equation.

Algebraic Manipulation

Algebraic manipulation is a critical skill in solving equations. It involves rearranging and simplifying expressions using basic arithmetic operations and algebraic rules.
To solve our linear equations$2x+7 = 5$and$2x+7 = -5$, you need to manipulate them as follows:

First, subtract $7$ from both sides in each equation. This isolates the term with $x$:
For $2x+7=5$, calculate $2x = 5 - 7$, leading to $2x = -2$.
For $2x+7=-5$, calculate $2x = -5 - 7$, leading to $2x = -12$.

Next, divide each resulting equation by $2$ to solve for $x$:

$x = -1$ for the first equation
$x = -6$ for the second

By learning these algebraic manipulations, solving both simple and complex equations becomes much more manageable.

Problem 94

Other exercises in this chapter

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Problem 94

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Problem 94

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Problem 94

Solve each absolute value inequality. $$\left|2-\frac{x}{2}\right|-1 \leq 1$$

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