Problem 20 - Elementary Algebra

Question

For the following problems, solve the equations by completing the square. $$ x^{2}-10 x=0 $$

Step-by-Step Solution

Answer

Question: Solve the quadratic equation by completing the square: \(x^2 - 10x = 0\). Answer: The solutions to the given equation are \(x = 0\) and \(x = 10\).

Step 1: Move the constant term to the right-hand side

We need to have the equation in the form \((x-a)^2=b\), so let's first move the constant term to the right-hand side of the equation. In this case, the constant term is already on the right side, so we have: $$ x^2 - 10x = 0 $$

Step 2: Add the square of half of the coefficient of x on both sides to complete the square

Now, we'll add the square of half of the coefficient of \(x\) to both sides of the equation, in order to complete the square: $$ x^2 - 10x + (-\frac{10}{2})^2 = 0 + (-\frac{10}{2})^2 $$ Simplify: $$ x^2 - 10x + 25 = 25 $$

Step 3: Rewrite the left-hand side as a perfect square

We can rewrite the left-hand side of the equation as a perfect square of the form \((x-a)^2\): $$ (x - 5)^2 = 25 $$

Step 4: Solve for x

Now, we can solve for \(x\) by taking the square root of both sides: $$ \sqrt{(x - 5)^2} = \sqrt{25} $$ The result is: $$ x - 5 = \pm 5 $$ So, we have two possible solutions: $$ x - 5 = 5 \Rightarrow x = 10 $$ and $$ x - 5 = -5 \Rightarrow x = 0 $$ Thus, the solutions to the given equation are \(x = 0\) and \(x = 10\).

Key Concepts

Quadratic Equations

A quadratic equation is a type of polynomial equation of degree 2. It is commonly expressed in the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. The term \( ax^2 \) is what gives the equation its quadratic nature because it involves the square of the variable \( x \). Quadratic equations are fundamental in algebra and have many applications in science and engineering. Anytime you encounter an equation involving \( x^2 \), you are likely dealing with a quadratic equation.
Solving quadratic equations can be approached through different methods, such as factoring, using the quadratic formula, or completing the square. Each method has its own advantages, depending on the specific characteristics of the equation at hand.

• Factoring: Suitable when the quadratic can be easily expressed as a product of binomials.
• The Quadratic Formula: Useful for any quadratic equation, especially when factoring is not straightforward.
• Completing the Square: Helps in understanding the properties of the parabola described by the equation.

Understanding these methods will allow you to solve any quadratic equation efficiently.

Algebraic Method

An algebraic method refers to a technique in algebra used to find solutions to equations or simplify expressions. Completing the square is one such powerful algebraic method often used to solve quadratic equations. By transforming a quadratic equation into a perfect square trinomial, this method simplifies the process of finding the solutions.
The idea is to manipulate the equation into the form \((x-a)^2 = b\), which makes it much easier to solve for \(x\). Here's how it typically works:

• Isolate the quadratic and linear terms on one side of the equation.
• Find the value that completes the square, which is \( \left( \frac{b}{2} \right)^2 \).
• Add this value to both sides of the equation to balance it.
• Rewrite the left-hand side as a perfect square trinomial.

By using this method, you gain a deeper insight into the configuration of the equation, and it ultimately becomes a stepping stone to more advanced algebraic concepts.

Solving Equations

Solving equations is a fundamental skill in algebra that involves finding the values of the variable that make the equation true. Quadratic equations, in particular, can be solved by various methods, and completing the square is a particularly insightful one.
In the exercise "\(x^2 - 10x = 0\)", completing the square involves a few key steps:

• First, recognize that there's no constant term, simplifying the completion process.
• Add the square of half the \(x\) coefficient to both sides, i.e., \( \left( \frac{-10}{2} \right)^2 = 25 \).
• Rewrite the equation as \((x-5)^2 = 25\).
• Take the square root of both sides, giving two potential solutions: \(x - 5 = \pm 5\).

From \(x - 5 = 5\) and \(x - 5 = -5\), solve to get \(x = 10\) and \(x = 0\), which are the solutions to the equation. These steps illustrate the beauty and simplicity of solving equations through symmetry and balance.