Problem 50 - Elementary Algebra

Question

For the following problems, solve each of the quadratic equations using the method of extraction of roots. $$ (x-3)^{2}=25 $$

Step-by-Step Solution

Answer

Answer: The solutions for the equation are x = 8 and x = -2.

Step 1: Taking the square root of both sides

The equation we've been given is: $$(x-3)^2=25$$ We start by taking the square root of both sides in order to isolate x: $$\sqrt{(x-3)^2}=\sqrt{25}$$

Step 2: Simplify the equation

Now we can simplify both sides of the equation. The square root of \((x-3)^2\) is \(x-3\) because \((x-3)^2\) means that \((x-3)\) is multiplied by itself. The square root of 25 is 5 and -5, because both 5 and -5 when squared give us 25. Therefore, the simplified equation is: $$x-3=\pm5$$

Step 3: Solve for x

Now, we can solve for x in the equation \(x-3=\pm5\). Depending on whether we take the positive or negative value of 5, we will have two different solutions for x. First, if we take the positive value of 5: $$x-3=5$$ $$x=5+3$$ $$x=8$$ Next, if we take the negative value of 5: $$x-3=-5$$ $$x=-5+3$$ $$x=-2$$ Thus, we have two solutions for the given quadratic equation: $$x=8, -2$$

Key Concepts

extraction of roots

Extraction of roots is a useful method when solving quadratic equations that are written in a certain form. A quadratic equation can sometimes be expressed as the square of a binomial expression set equal to a number. For example, let's consider the equation \((x-3)^2 = 25\).
To solve such equations using extraction of roots, the first step involves taking the square root of both sides of the equation.
This helps in simplifying the expression, leading towards finding the value of the variable.
By taking the square root of \((x-3)^2\), we end up with \(x-3\).
The right side, \(25\), simplifies to \(5\) or \(-5\) since the square of both 5 and -5 is 25.
Remember that whenever we take the square root as part of solving an equation, we must consider both the positive and negative solutions.
In summary, extraction of roots effectively reduces a squared term and unveils both potential solutions of the quadratic equation.

solving quadratic equations

Quadratic equations come in the form \(ax^2 + bx + c = 0\) and can be solved using various methods. The problem we are looking at is a simplified one where the equation is easily factored into a binomial form.
In our example problem, \((x-3)^2 = 25\), we do not have to use more complex methods like factoring or the quadratic formula. Instead, we use extraction of roots for a swift solution.
Here's how you follow through:

• First, take the square root of both sides, isolating the expression and simplifying the equation.
• We'll discover two potential values for \((x-3)\), namely 5 and -5.
• Finally, solve each resulting simple equation \(x-3 = 5\) and \(x-3 = -5\) to find that \(x = 8\) and \(x = -2\).

This showcases that quadratic equations can often yield two different solutions, considering their parabolic shape. Understanding how to manipulate and solve these equations is key in algebra.

square roots

Square roots are an essential mathematical concept that often crop up in solving quadratic equations.
When you take the square root of a number, you are looking for a value that, when multiplied by itself, gives the original number.
In algebra, the result of a square root operation can be both positive and negative. For instance, the square root of 25 is both \(5\) and \(-5\); both values squared return to 25.
This dual possibility plays a critical role in finding solutions to quadratic equations, as seen in the equation \((x-3)^2 = 25\).
Square roots are represented mathematically by the radical symbol, \(\sqrt{}\).
When applied to equations, taking the square root on both sides helps simplify the form and clarify the solution route.

• Always remember that solving involves both potential roots: + and -.
• It's crucial to apply this operation correctly to avoid missing solutions.

Thus, understanding the fundamental properties of square roots is vital for successfully navigating quadratic equations.