Textbook Question
Solve each equation in Exercises 15–34 by the square root property.(x + 3)^2 = - 16
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1. Equations & Inequalities
The Square Root Property
3:09 minutesProblem 32
Textbook Question
Solve each equation using the square root property. See Example 2. (4x + 1)^2 = 20
Verified step by step guidance1
Start by isolating the squared term. The equation is already in the form \((4x + 1)^2 = 20\).
Apply the square root property, which states that if \(a^2 = b\), then \(a = \pm \sqrt{b}\). So, take the square root of both sides: \(4x + 1 = \pm \sqrt{20}\).
Simplify \(\sqrt{20}\) if possible. \(\sqrt{20} = \sqrt{4 \times 5} = \sqrt{4} \times \sqrt{5} = 2\sqrt{5}\).
Now, rewrite the equation as \(4x + 1 = \pm 2\sqrt{5}\).
Solve for \(x\) by first subtracting 1 from both sides: \(4x = -1 \pm 2\sqrt{5}\), and then divide each term by 4: \(x = \frac{-1 \pm 2\sqrt{5}}{4}\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Square Root Property
The square root property states that if a squared expression equals a number, then the original expression can be solved by taking the square root of both sides. Specifically, if (a)^2 = b, then a = ±√b. This property is essential for solving quadratic equations and allows for finding the values of the variable involved.
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Isolating the Squared Term
To effectively use the square root property, it is crucial to isolate the squared term on one side of the equation. In the given equation (4x + 1)^2 = 20, we first ensure that (4x + 1)^2 is alone on one side before applying the square root property. This step is vital for correctly applying the property and finding the correct solutions.
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Handling ± Solutions
When applying the square root property, it is important to remember that taking the square root of a number yields both a positive and a negative solution. For example, if we find that (4x + 1) = ±√20, we must consider both cases: 4x + 1 = √20 and 4x + 1 = -√20. This ensures that all possible solutions to the equation are accounted for.
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