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Chapter 8: Problem 31

Use the square root property to solve each equation. \((x+2)^{2}=25\)

Short Answer

Expert verified
The solutions are \(x = 3\) and \(x = -7\).

Step by step solution

01

Isolate the squared term

The equation is already in the form of \((x + 2)^{2} = 25\), where the squared expression \(x + 2\) is isolated.
02

Apply the square root property

Apply the square root property to both sides of the equation. The square root property states that if \((x + 2)^{2} = 25\), then \[x + 2 = \pm \sqrt{25}.\]
03

Simplify the square root

Simplify the square root of 25, \(\root 25 = 5\), to get two possible equations: \(x + 2 = 5\) and \(x + 2 = -5\).
04

Solve for x

Solve each equation for \(x\). \((x + 2 = 5)\) gives \(x = 5 - 2 = 3\). \((x + 2 = -5)\) gives \(x = -5 - 2 = -7\).

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Key Concepts

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

square root property
To solve quadratic equations, the square root property is often very useful. This property states that if we have an equation like \[((x + a)^{2} = b)\], we can take the square root of both sides to solve for \[x + a\]. This gives us \[x + a = \pm \sqrt{b}.\] This step is foundational because it simplifies our equation significantly, bringing it closer to isolating the variable. Essentially, it breaks down a complex quadratic equation into two simpler linear equations.
isolating the variable
Isolating the variable is a key step in solving equations. It means getting the variable, usually \[x\], alone on one side of the equation. In our example, the variable term \[((x + 2)^{2})\] is already isolated on one side. This is ideal, as it allows us to apply the square root property directly. In other cases, you might have to perform additional operations, like adding or subtracting terms, to get the variable on one side by itself before proceeding.
simplifying square roots
Once you apply the square root property, the next step is to simplify the square root. Simplification means finding the principal value of the square root of the number. In our example, the number under the square root is \[25\], and its square root is \[5\]. This step gives us two linear equations: \[x + 2 = 5\] and \[x + 2 = -5\]. No matter what number is under the square root, you'll follow a similar process of finding its positive and negative square roots.
finding solutions to equations
Finally, solving the simplified linear equations will give us the solutions to the original quadratic equation. For \[x + 2 = 5\], you subtract 2 from both sides to get \[x = 3\]. For \[x + 2 = -5\], you do the same subtraction to get \[x = -7\]. So the solutions to the equation \[\((x+2)^{2}=25\)\] are \[x=3\] and \[x=-7\]. Always remember to check your solutions by plugging them back into the original equation to verify they work correctly.

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Most popular questions from this chapter

Solve each problem using a quadratic equation. A certain bakery has found that the daily demand for blueberry muffins is \(\frac{6000}{p},\) where \(p\) is the price of a muffin in cents. The daily supply is \(3 p-410 .\) Find the price at which supply and demand are equal.

Solve each inequality, and graph the solution set. $$ \frac{3 x+7}{x-3} \leq 0 $$

Solve each equation. Check the solutions. $$3-\frac{7}{2 p+2}=\frac{6}{(2 p+2)^{2}}$$

A club swimming pool is \(30 \mathrm{ft}\) wide and \(40 \mathrm{ft}\) long. The club members want an exposed aggregate border in a strip of uniform width around the pool. They have enough material for \(296 \mathrm{ft}^{2}\). How wide can the strip be?

Solve each inequality, and graph the solution set. $$ \frac{2 x-3}{x^{2}+1} \geq 0 $$
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