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Chapter 10: Problem 11

Solve the following quadratic equations. \(\frac{2}{5} a^{2}+3=11\)

Short Answer

Expert verified
The solutions are \(a = 2\sqrt{5}\) and \(a = -2\sqrt{5}\).

Step by step solution

01

Isolate the quadratic term

Subtract 3 from both sides of the equation to isolate the quadratic term: \[ \frac{2}{5} a^{2} + 3 - 3 = 11 - 3 \] Simplifying, we get: \[ \frac{2}{5} a^{2} = 8 \]
02

Eliminate the fraction

Multiply both sides of the equation by 5 to eliminate the fraction: \[ 5 \cdot \frac{2}{5} a^{2} = 8 \cdot 5 \] Simplifying, we get: \[ 2a^{2} = 40 \]
03

Solve for \(a^{2}\)

Divide both sides of the equation by 2 to solve for \(a^{2}\): \[ \frac{2a^{2}}{2} = \frac{40}{2} \] Simplifying, we get: \[ a^{2} = 20 \]
04

Solve for \(a\)

Take the square root of both sides to solve for \(a\): \[ a = \pm \sqrt{20} \] Simplifying, we get: \[ a = \pm 2\sqrt{5} \]

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Key Concepts

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

quadratic equations
Quadratic equations are essential in algebra and a cornerstone for understanding higher mathematics. They generally take the form: domains: \[ ax^2 + bx + c = 0 \] Here, 'a', 'b', and 'c' are constants. The highest exponent is squared. This is why they are called 'quadratic'. Quadratic equations can be solved using various methods like factoring, completing the square, or using the quadratic formula. The method we focus on here involves isolating the variable and simplifying step by step.
Understanding these basics makes solving quadratic equations easier. Let's dive into the details of solving them.
isolate the variable
To solve an equation, you need to isolate the unknown variable. This means getting the variable alone on one side of the equation. Let's take a look at the given example: domains: \[ \frac{2}{5} a^2 + 3 = 11 \] We aim to have 'a' alone on one side. First, subtract 3 from both sides: \[ \frac{2}{5} a^2 + 3 - 3 = 11 - 3 \] This gives us: \[ \frac{2}{5} a^2 = 8 \] By eliminating numbers around the variable step by step, we are closer to isolating 'a'. This makes solving the equation straightforward.
square root method
The square root method is a great tool for solving quadratic equations, especially when the equation takes the form \[ x^2 = n \]Once the quadratic term is isolated, we can apply the square root method. In the example, we reached: \[ a^2 = 20 \] Taking the square root of both sides, we get: \[ a = \pm \sqrt{20} \]Remember, you get both positive and negative roots as solutions in quadratic equations. Simplifying further, we can write: \[ a = \pm 2\sqrt{5} \] This method is effective when dealing with quadratic terms. Always include both positive and negative roots in your final answer.
simplifying fractions
Fractions can make equations look complex but simplifying them is often straightforward. In our example: \[ \frac{2}{5} a^2 = 8 \]Multiplying both sides by 5 eliminates the fraction: \[ 5 \cdot \frac{2}{5} a^2 = 8 \cdot 5 \] Simplifying, we get: \[ 2a^2 = 40 \]Next, dividing both sides by 2 gives: \[ a^2 = 20 \]This shows how simplifying fractions can make equations easier to deal with. Treat each fractional term carefully and work to clear them out at the earliest steps.

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Most popular questions from this chapter

In the following exercises, solve. Round answers to the nearest tenth. A retailer who sells backpacks estimates that, by selling them for \(x\) dollars each, he will be able to sell \(100-x\) backpacks a month. The \(\quad\) quadratic \(\quad\) equation \(R=-x^{2}+100 x\) is used to find the \(R\) received when the selling price of a backpack is \(x\). Find the selling price that will give him the maximum revenue, and then find the amount of the maximum revenue.

Solve by using the Quadratic Formula. \(p^{2}-6 p-27=0\)

In the following exercises, graph by using intercepts, the vertex, and the axis of symmetry. $$ y=-x^{2}+2 x-7 $$

Solve by using the Quadratic Formula. \(v(v+5)-10=0\)

In the following exercises, graph by using intercepts, the vertex, and the axis of symmetry. $$ y=x^{2}+4 x-12 $$
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