Problem 81 - Elementary Algebra

Question

Solve by completing the square. \(a^{2}-10 a=-5\)

Step-by-Step Solution

Answer

The solutions are \( a = 5 + 2\sqrt{5} \) and \( a = 5 - 2\sqrt{5} \).

Step 1: Move the constant term to the right side

Start by adding 5 to both sides to isolate the quadratic and linear terms on the left side: \[ a^{2} - 10a = -5 \] Adding 5 to both sides:\[ a^{2} - 10a + 5 \rightarrow 0 \] So, the equation is now: \[ a^{2} - 10a + 5 = 0 \]

Step 2: Add and subtract a specific value to complete the square

To complete the square, take half the coefficient of the linear term (-10), square it, and add it to both sides.Half of -10 is -5, and (-5) squared is 25. Add and subtract 25: \[ a^{2} - 10a + 25 - 25 + 5 = 0 \]

Step 3: Rearrange and simplify

Combine and simplify the terms inside the parentheses:\[ (a - 5)^2 - 25 + 5 = 0 \]Simplify further: \[ (a - 5)^2 - 20 = 0 \]

Step 4: Isolate the square term

Add 20 to both sides to isolate the squared term:\[ (a - 5)^2 = 20 \]

Step 5: Solve for a

Take the square root of both sides to solve for a: \[ a - 5 = \pm\sqrt{20} \]Remember that \sqrt{20} simplifies to 2\sqrt{5}.So, \[ a - 5 = \pm 2\sqrt{5} \]Finally, add 5 to both sides to solve for a:\[ a = 5 \pm 2\sqrt{5} \]

Key Concepts

Quadratic Equations

Quadratic equations are equations of the form

ax^2 + bx + c = 0

where a, b, and c are constants.
The solutions to a quadratic equation are the values of x that make the equation true.
Examples of quadratic equations are 2x^2 + 3x - 5 = 0 and x^2 - 4x + 4 = 0.
There are various methods to solve quadratic equations:

• Factoring
• Using the quadratic formula
• Graphing
• Completing the square

In our exercise, we use the completing the square method.

Solving Equations

Solving equations involves finding the value of the variable(s) that make the equation true.
There are different types of equations, like linear, quadratic, and polynomial.
In solving quadratic equations, the goal is to isolate the variable.
For our exercise, we follow specific steps:

• Move constant terms to one side
• Complete the square
• Isolate the variable

By moving terms around, combining like terms, and manipulating the equation, we solve for the variable.
Each step focuses on making the equation simpler and isolating the unknown variable.

Algebraic Manipulation

Algebraic manipulation involves rearranging equations to make them easier to solve.
This includes combining like terms, using arithmetic operations, and applying algebraic identities.
In the context of completing the square:

• Move constants to isolate the quadratic and linear terms
• Add and subtract specific values to form a perfect square trinomial
• Factor the trinomial
• Isolate the squared term

These steps require understanding how to balance equations and manipulate them while maintaining equality.