Problem 17 - Elementary Algebra

Question

Solve by completing the square. $$ x 2-12 x+1=0 $$

Step-by-Step Solution

Answer

The solutions are \( x = 6 + \sqrt{35} \) and \( x = 6 - \sqrt{35} \).

Step 1: Rewrite the Equation

Start with the given equation: \( x^2 - 12x + 1 = 0 \).The quadratic is in the form \( ax^2 + bx + c \), which is already set with \( a = 1 \), \( b = -12 \), and \( c = 1 \).

Step 2: Move the Constant Term to the Other Side

Subtract the constant term \( c = 1 \) from both sides:\[ x^2 - 12x = -1 \].

Step 3: Complete the Square

To complete the square, take \( b/2 \) and square it:\[ \left(-12/2\right)^2 = (-6)^2 = 36 \].Add and subtract this square inside the equation:\[ x^2 - 12x + 36 = 35 \].

Step 4: Rewrite the Quadratic as a Perfect Square

Write the left side of the equation as a square of a binomial:\[ (x - 6)^2 = 35 \].

Step 5: Solve for x

Take the square root of both sides:\[ \sqrt{(x - 6)^2} = \pm\sqrt{35} \],which gives\[ x - 6 = \pm\sqrt{35} \].Solve for \( x \) by adding 6 to both sides:\[ x = 6 \pm \sqrt{35} \].

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation of degree 2. These equations are usually in the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( x \) represents an unknown variable.
Quadratic equations are crucial in algebra as they describe a wide range of phenomena, from physical movements to financial calculations.
Quadratics can be visualized as parabolic graphs on a coordinate plane, where the graph's shape and position depend on the values of \( a \), \( b \), and \( c \).
If \( a = 0 \), the equation becomes linear, which simplifies the problem significantly. However, since \( a eq 0 \) in our case, the equation remains quadratic.

Perfect Square

A perfect square in mathematics refers to an expression that is the square of a binomial. Essentially, for any binomial expression \( (x + d) \), its square is \((x + d)^2\), which expands to \(x^2 + 2dx + d^2\).
In the context of completing the square, we transform a quadratic equation into a perfect square trinomial.
For example, starting with the equation \( x^2 - 12x \), we can complete the square by finding a number that turns it into a perfect square trinomial.
We do this by halving the coefficient of \( x \), squaring this result, and adding it to both sides of the equation.

• Here, half of \(-12\) is \(-6\).
• Squaring \(-6\) gives us \( 36 \), which is added to both sides.

This transformation allows us to rewrite the equation in the simple form \((x - 6)^2\), making it easier to solve.

Solving Equations

Solving a quadratic equation can be achieved through various methods, one of which is completing the square. This technique is advantageous when the quadratic cannot be factored easily.
Completing the square involves transforming the quadratic part of the equation into a perfect square trinomial.
In our exercise, after transforming \( x^2 - 12x + 36 = 35 \) into \( (x - 6)^2 = 35 \), we proceed to extract the square root of both sides.
This gives us two potential solutions since squaring a number can yield a positive or negative result:

• \( x - 6 = \sqrt{35} \)
• \( x - 6 = -\sqrt{35} \)

By isolating \( x \), we determine the two solutions \( x = 6 + \sqrt{35} \) and \( x = 6 - \sqrt{35} \).
This dual solution arises because a quadratic function is symmetric and crosses the x-axis at two points.

Step by Step Solution

A step-by-step solution for solving equations using the completing the square method can be invaluable for learners. It breaks down each part of the process methodically.

• Step 1 involves ensuring the quadratic equation is in standard form.
• Step 2 requires moving the constant term to the other side of the equation.
• Step 3 is where you "complete the square." Calculate \((b/2)^2\) and add it to both sides to form a perfect square trinomial.

This precision helps to encapsulate the quadratic as \((x - d)^2\).

• Step 4 is rewriting the trinomial as a perfect square.
• Step 5 involves solving for \( x \) by taking the square root of both sides and finding both potential solutions of the quadratic equation.

By methodically following these steps, even complex quadratic equations become manageable, offering clarity and understanding of how the solution is derived.