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Chapter 8: Problem 27

Use the square root property to solve each equation. \(2 x^{2}+7=61\)

Short Answer

Expert verified
x = \pm 3\sqrt{3}

Step by step solution

01

Isolate the quadratic term

Start by isolating the term that contains the variable. Subtract 7 from both sides of the equation to get: \[ 2x^2 + 7 - 7 = 61 - 7 \] Which simplifies to: \[ 2x^2 = 54 \]
02

Divide both sides by the coefficient of the quadratic term

To isolate \(x^2\), divide both sides of the equation by 2: \[ \frac{2x^2}{2} = \frac{54}{2} \] Simplifying that, we get: \[ x^2 = 27 \]
03

Apply the square root property

To solve for \(x\), take the square root of both sides. Remember to include both the positive and negative solutions: \[ x = \pm \sqrt{27} \] Simplify the square root of 27: \[ x = \pm 3\sqrt{3} \]

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Key Concepts

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

Square Root Property
The square root property is an essential tool in solving quadratic equations. Simply put, if a quadratic equation can be written in the form \(x^2 = k\), then the solutions to the equation are \(x = \pm \sqrt{k}\). This property is useful when the quadratic term is isolated on one side of the equation.
In our example, once we have \(x^2 = 27\), we can apply the square root property:
\[ x = \pm \sqrt{27} \]
Always remember to include both the positive and negative roots when applying this property.
Let's move on to learn how to isolate the quadratic term, which is often the first crucial step.
Isolating Quadratic Term
When solving a quadratic equation, it's crucial to isolate the quadratic term (the term involving \(x^2\)) first. In the given problem, the original equation is \(2x^2 + 7 = 61\).
Here's how to isolate the quadratic term step-by-step:
  • First, subtract 7 from both sides of the equation:
    \[ 2x^2 + 7 - 7 = 61 - 7 \]
  • This simplifies to:\br> \[ 2x^2 = 54 \]
  • Next, divide both sides by 2 (the coefficient of \(x^2\)):
    \[ \frac{2x^2}{2} = \frac{54}{2} \]
  • Simplify this to get:
    \[ x^2 = 27 \]
Now, with \(x^2\) isolated, we are ready to apply the square root property and solve for \(x\).
Simplifying Square Roots
The final step in solving our quadratic equation is to simplify the square root. After applying the square root property, we got:
\[ x = \pm \sqrt{27} \]

Square roots often need simplifying. The square root of 27 can be broken down into its prime factors:
\[ \sqrt{27} = \sqrt{3 \cdot 3 \cdot 3} \]

We pair the 3s inside the square root to simplify:
\br> \[ x = \pm 3 \sqrt{3} \]

Hence, the simplified solutions for our equation \(2x^2 + 7 = 61\) are \( x = 3 \sqrt{3} \) and \( x = -3 \sqrt{3} \). Remember, it鈥檚 always essential to simplify square roots to their simplest form.

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