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Chapter 10: Problem 1

Solve the following quadratic equations. \(a^{2}=49\)

Short Answer

Expert verified
The solutions are \(a = 7\) and \(a = -7\).

Step by step solution

01

Identify the Quadratic Equation

The given equation is \(a^{2} = 49\). This is already in a simple quadratic form.
02

Take the Square Root of Both Sides

To solve for \(a\), take the square root of both sides of the equation. Remember that taking the square root of both sides will give two possible solutions (positive and negative): \[a = \pm \sqrt{49}\].
03

Simplify the Square Root

Simplify \(\sqrt{49}\). Since \(49\) is a perfect square, \(\sqrt{49} = 7\). Thus, \[a = \pm 7\].
04

Write the Solution Set

Combine the results to write the final solution set. The values of \(a\) are \[a = 7 \text{ and } a = -7\].

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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 polynomial equations of degree 2. This means they involve an unknown variable raised to the power of 2.
A general form of a quadratic equation is given by: \[ax^2 + bx + c = 0\]Here, 'a', 'b', and 'c' are constants, and 'x' is the variable.
These equations often describe parabolic shapes when plotted on a graph.
To solve them, we use various methods such as:
  • Factoring
  • Using the quadratic formula \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
  • Completing the square
  • Graphing
Each method has its own use cases and advantages. For our problem \(a^2 = 49\), since it's already simplified, taking the square root is the best approach.
Square Root
The square root of a number is a value that, when multiplied by itself, gives the number. For instance, the square root of 49 is 7 because \(7^2 = 49\).
In mathematical notation, the square root of 'x' is represented as \(\sqrt{x}\).
When solving \(a^2 = 49\), we take the square root of both sides to isolate 'a': \[a = \pm\sqrt{49}\] It's crucial to remember that there are always two square roots for any positive number: one positive and one negative. This is why we write \(\pm\) before the square root value, meaning \(a = 7\) and \(a = -7\). These two values are the solutions to the quadratic equation in our example.
Solution Set
In mathematics, a solution set is the set of all values that satisfy a given equation. In the context of quadratic equations, this means the values of 'x' (or any variable) that make the equation true.
For our specific problem \(a^2 = 49\), after taking the square root of both sides we get two solutions \(a = 7\) and \(a = -7\).
So, the solution set is \[a = \{7, -7\}\].
Always be sure to include all possible values in the solution set, as leaving out any valid solutions can result in an incomplete answer.

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Most popular questions from this chapter

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