Problem 36 - Elementary Algebra

Question

For the following problems, solve each of the quadratic equations using the method of extraction of roots. $$ 4 a^{2}=40 $$

Step-by-Step Solution

Answer

Answer: The two solutions are $$a = \sqrt{10}$$ and $$a = -\sqrt{10}$$.

Step 1: Isolate the term with the variable

Divide both sides of the equation by 4 to isolate the term $$a^2$$: $$ \frac{4a^2}{4} = \frac{40}{4} \\ a^2 = 10 $$

Step 2: Use the extraction of roots method

Take the square root of both sides of the equation to solve for $$a$$: $$ \sqrt{a^2} = \sqrt{10} \\ a = \pm\sqrt{10} $$

Step 3: Write the final answer

The solutions to the quadratic equation $$4a^2 = 40$$ are $$a = \sqrt{10}$$ and $$a = -\sqrt{10}$$.

Key Concepts

Quadratic Equations

Quadratic equations are mathematical expressions that take the form of \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(x\) represents the variable or unknown that we aim to solve for. These equations are called 'quadratic' because the highest degree of the variable \(x\) is two, known as a square. The solutions to a quadratic equation are also referred to as the 'roots' of the equation.

One of the methods for solving these equations is the extraction of roots, also known as 'taking the square root' of both sides. This method is particularly handy when the quadratic equation is already in the form of \(x^2 = n\) and can be applied easily as seen in our example \(4a^2 = 40\). This method leads us to identify two potential solutions, as both a positive and a negative number squared will yield the same positive result. Therefore, any time you apply the square root to both sides of the equation, remember to consider both the positive and negative solutions.

Isolating Variables

Isolating the variable in an equation is a fundamental skill in algebra, crucial for solving equations effectively. The goal is to manipulate the equation so that the variable we want to solve for stands alone on one side of the equal sign, giving us a clear view of its value. To isolate a variable, we can perform operations like addition, subtraction, multiplication, division, and taking roots, as long as we apply them to both sides of the equation to maintain balance.

Looking at our example, by dividing both sides by \(4\), we isolated \(a^2\) to one side, making the step towards solving for \(a\) much clearer. It's important to understand the logic behind each operation we choose, as they help us simplify the equation step by step until we can plainly see the solution or, in the case of quadratic equations, solutions.

Square Roots

Square roots are a fundamental concept in mathematics, associated with the operation of finding a number that, when multiplied by itself, gives the initial non-negative number we started with. The square root of a number \(n\) is denoted as \(\sqrt{n}\). In the context of quadratic equations, once a variable squared is isolated on one side (\(x^2 = n\)), taking the square root of both sides is the next logical step to deduce the value of the variable.

However, it's important to be aware of the fact that every positive number has two square roots: one positive and one negative, since \(\left(-x\right)^2 = x^2\). This gives us two possible solutions for the variable when solving quadratic equations. In the example given, \(\sqrt{10}\) and \(\-\sqrt{10}\) are both valid, since squaring either value results in \(10\). Always remember to include both the positive and negative roots in your final answer.