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Chapter 8: Problem 30

Use the square root property to solve each equation. \(4 x^{2}-7=65\)

Short Answer

Expert verified
The solutions are \(x = 3\sqrt{2}\) and \(x = -3\sqrt{2}\).

Step by step solution

01

Isolate the quadratic term

First, isolate the term that includes the square. In the given equation \(4x^{2} - 7 = 65\), add 7 to both sides to get:\(4x^{2} = 72\).
02

Divide by the coefficient of the quadratic term

Next, divide both sides of the equation by 4, the coefficient of the quadratic term: \(\frac{4x^{2}}{4} = \frac{72}{4}\), which simplifies to: \(x^{2} = 18\).
03

Apply the square root property

Apply the square root property to solve for \(x\). Taking the square root of both sides gives us two solutions: \(x = \pm \sqrt{18}\).
04

Simplify the square root

Simplify the square root of 18. Since \(\sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2}\), the solutions are \(x = \pm 3\sqrt{2}\).

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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 a very useful tool for solving quadratic equations. When you have an equation in the form of (a variable squared equals some number), you can solve it by taking the square root of both sides. Remember, when you take the square root of both sides, you must include both the positive and negative roots. This gives you two possible solutions. For example, let's consider the equation: \[x^{2} = 18\] Taking the square root of both sides, we have: \[x = \pm\sqrt{18}\] So, using the square root property, we always remember to write \pm (plus and minus) to indicate both positive and negative solutions.
isolating the quadratic term
When solving a quadratic equation using the square root property, the first step is to isolate the quadratic term. This means you want to get the term with the variable squared by itself on one side of the equation. Let's look at the example equation: \[4x^{2} - 7 = 65\] To isolate the quadratic term \(4x^{2}\), start by moving the constant (-7) to the other side of the equation: \[4x^{2} - 7 + 7 = 65 + 7\] This simplifies to: \[4x^{2} = 72\] At this point, the quadratic term \(4x^{2}\) is isolated, and we are ready for the next step.
simplifying square roots
Simplifying square roots is an important step when solving quadratic equations, especially after applying the square root property. It involves breaking down the square root into simpler terms. Consider our example: \[x = \pm\sqrt{18}\] To simplify \(\sqrt{18}\), we factor it into its prime factors: \[\sqrt{18} = \sqrt{9 \cdot 2}\] Since \(9\) is a perfect square, we can simplify further: \[\sqrt{18} = \sqrt{9} \cdot \sqrt{2} = 3\sqrt{2}\] Therefore, the solutions become: \[x = \pm 3\sqrt{2}\]
quadratic term
The quadratic term in an equation is the term that includes the variable raised to the second power (squared). In the equation \[4x^{2} - 7 = 65\] \(4x^{2}\) is the quadratic term. Properly handling the quadratic term is crucial for solving the equation efficiently.
  • First, isolate it.
  • Then, manage the coefficients correctly.
  • Finally, apply appropriate properties, like the square root property.
By focusing on the quadratic term correctly, you make the problem simpler to solve step-by-step.

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Most popular questions from this chapter

A rectangle has a length \(2 \mathrm{~m}\) less than twice its width. When \(5 \mathrm{~m}\) are added to the width, the resulting figure is a square with an area of \(144 \mathrm{~m}^{2}\). Find the dimensions of the original rectangle.

The following exercises are not grouped by type. Solve each equation. $$\left(x^{2}+x\right)^{2}+12=8\left(x^{2}+x\right)$$

In the 1939 classic movie The Wizard of Oz, Ray Bolger's character, the Scarecrow, wants a brain. When the Wizard grants him his "Th.D." (Doctor of Thinkology), the Scarecrow replies with the following statement. Scarecrow: The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side. His statement sounds like the formula for the Pythagorean theorem. Use wording similar to that of the Scarecrow, but state the Pythagorean theorem correctly.

It takes \(m\) hours to grade a set of papers. (a) What is the grader's rate (in job per hour)? (b) How much of the job will the grader do in \(2 \mathrm{hr}\) ?

Solve each inequality, and graph the solution set. $$ \frac{x-8}{x-4}<3 $$
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