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Chapter 11: Problem 21

In Exercises \(1-64\), solve each of the given equations. If the equation is quadratic, use the factoring or square root method. If the equation has no real solutions, say so.$$9 a^{2}=20$$

Short Answer

Expert verified
The solutions are \(a = \pm \frac{2\sqrt{5}}{3}\)

Step by step solution

01

Isolate the quadratic term

Start by isolating the quadratic term on one side of the equation. The given equation is \[9a^{2} = 20\]
02

Divide by the coefficient of the quadratic term

Divide both sides of the equation by 9 to solve for \(a^2\):\[a^2 = \frac{20}{9}\]
03

Apply the square root method

Take the square root of both sides of the equation to solve for \(a\):\[a = \pm \sqrt{\frac{20}{9}}\]
04

Simplify the square root

Simplify the square root of the fraction:\[a = \pm \frac{\sqrt{20}}{3}\]
05

Further simplify the square root

Simplify \(\sqrt{20}\) to its simplest form:\[\sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5}\]Thus:\[a = \pm \frac{2\sqrt{5}}{3}\]

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Key Concepts

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

factoring method
The factoring method involves expressing a quadratic equation in a form where it can be factored into simpler expressions multiplied together, typically in the form \( ax^2 + bx + c = 0 \). After factoring, you set each factor to zero and solve for the variable. While the original exercise doesn't use this method, it's important to understand it for other quadratic equations, especially those that are not easily simplified by taking square roots.
Here鈥檚 a quick overview:
  • Find two numbers that multiply to get the constant term (c) and add to get the coefficient of the middle term (b).
  • Rewrite the middle term using these two numbers.
  • Factor by grouping and solve for the variable.
This method is particularly powerful for equations with integer solutions.
square root method
The square root method is useful for solving quadratic equations wherein the equation can be formatted as \( ax^2 = k \), like in the original exercise \( 9a^2 = 20 \). Here's a step-by-step breakdown:
  • Isolate the quadratic term: Move all other terms to the opposite side of the equation.
  • Divide by the coefficient of the squared variable: This makes the equation in the form \( x^2 = k \).
  • Apply the square root to both sides: Remember, taking the square root introduces both positive and negative solutions, \( x = \pm \sqrt{k} \).
This method offers a direct way to find solutions but works best when the equation is simplified to the quadratic term and a constant.
simplifying square roots
Simplifying square roots is key in solving many quadratic equations and involves breaking down the square root expression into its simplest form. When you take the square root of a non-perfect square number, look for factors that are perfect squares.
Let's revisit the original exercise where we simplified \( a = \pm \frac{2\sqrt{5}}{3} \):
  • Identify factors of the number under the square root that are perfect squares. For \( \sqrt{20} \), this is \sqrt{4 \times 5} \>.
  • Rewrite the square root as the product of two separate roots: \sqrt{20} = \sqrt{4 \times 5} = \sqrt{4} \cdot \sqrt{5} \>.
  • Simplify the perfect square factor: Since \sqrt{4} = 2, this turns into \ 2\sqrt{5} \.
There's also value in learning to recognize non-square numbers and breaking them down for simplicity. This technique makes the overall process of solving equations more manageable.

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Most popular questions from this chapter

In Exercises \(85-94\), solve the equations using the square root method. Round off your answers to the nearest hundredth. $$x^{2}=7$$

$$\text { In Exercises } 97-100, \text { solve the system of equations algebraically.}$$ $$\left\\{\begin{array}{l} 6 x-4 y=9 \\ 3 x-2 y=2 \end{array}\right.$$

Sketch the graph of the given equation. Find the intercepts; approximate to the nearest tenth where necessary. $$y=x^{2}-4 x+2$$

In Exercises \(75-84\), round your answer to the nearest tenth where necessary. The length of a rectangle is 3 more than the width. If the area of the rectangle is 70 sq in., find the dimensions of the rectangle.

Simplify as completely as possible. (Assume \(x \geq 0 .)\) $$\frac{12}{\sqrt{5}-\sqrt{3}}$$
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