Problem 10 - Elementary Algebra

Question

For the following problems, solve the equations. $$ (a-5)^{2}=0 $$

Step-by-Step Solution

Answer

Answer: The solution to the equation is \(a=5\).

Step 1: Apply the square root

Apply the square root to both sides of the equation: $$\sqrt{(a-5)^2} = \sqrt{0}$$

Step 2: Simplify square roots

Simplify the square roots: $$a-5 = 0$$

Step 3: Solve for a

Add 5 to both sides of the equation to solve for a: $$a = 5$$ The solution for the given equation is \(a=5\).

Key Concepts

Quadratic Equations

Quadratic equations are a type of polynomial equation where the highest exponent of the variable is 2. These equations follow the standard form of \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are known numbers, with \( a \) not equal to zero. Quadratic equations can graphically represent a parabola, which is a symmetrical curve. In many real-world situations, quadratic equations model scenarios like projectile motion, area problems, and more.

Solving quadratic equations can be done using several methods including:

• Factoring: Express the equation as a product of its factors.
• Quadratic Formula: For any quadratic equation \( ax^2 + bx + c = 0 \), the solution can be found using \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
• Completing the Square: Rearrange the equation into a perfect square trinomial.
• Graphical Method: Plotting the equation and finding the roots where the graph intersects the x-axis.

In the given exercise, the quadratic equation is simplified since it is expressed directly as a square, \((a-5)^2 = 0\), which makes solving straightforward through square roots.

Square Roots

Square roots are mathematical operations that find the number which, when multiplied by itself, gives the original number. For example, the square root of 9 is 3, written as \( \sqrt{9} = 3 \), since \( 3 \times 3 = 9 \).

In the context of equations, using square roots is a method to simplify and solve expressions. For the exercise, we used the square root to simplify \((a-5)^2 = 0\), resulting in \( \sqrt{(a-5)^2} = 0\). This becomes \( a-5 = 0 \), showing how square roots help isolate variables.

Square roots are vital in many areas of mathematics and science because:

• They are essential in geometry, particularly in finding distances.
• They factor into solving quadratic equations and polynomial identities.
• They appear in real-world problems involving proportions, physics, and engineering.

Remember, a square root can have both a positive and negative solution, but in our case, since squaring is zero, there is the only non-negative solution.

Algebraic Solutions

Algebraic solutions involve solving equations using algebraic manipulations and operations. When dealing with equations like \((a-5)^2 = 0\), the goal is to isolate the variable. This often means performing inverse operations to cancel parts of the equation.

In the exercise, we use the concept of inverse operations:

• Taking the Square Root: Applied to both sides to "undo" the square on \((a-5)\).
• Solving for the Variable: Once simplified, we find \(a = 5\) by adding 5 to both sides.

Algebraic solutions require understanding how to balance the equation by maintaining equality, which means whatever operation you apply to one side must be applied to the other.

Key principles include:

• Identifying operations that cancel each other, like multiplication and division, addition and subtraction.
• Employing distributive, associative, and commutative properties as needed.
• Checking your solution to ensure it satisfies the original equation.

By following these algebraic steps, you gain control over the solving process, ensuring that the result is accurate and meaningful.