Problem 73 - Elementary Algebra

Question

Solve by completing the square and round off the solutions to the nearest hundredth. $$9 x(x-1)-2(2 x-1)=-4 x$$

Step-by-Step Solution

Answer

Solutions are approximately 1.69 and -0.13.

Step 1: Expand and simplify the equation

Start by expanding all terms in the equation:\[9x(x-1) - 2(2x-1) = -4x\]This becomes: \[9x^2 - 9x - 4x + 2 = 0\]Now, combine like terms:\[9x^2 - 13x + 2 = 0\]

Step 2: Move constant to right side

To complete the square, we need to move the constant term to the right side of the equation:\[9x^2 - 13x = -2\]

Step 3: Make the leading coefficient 1

Divide the entire equation by 9 to make the coefficient of \(x^2\) equal to 1:\[x^2 - \frac{13}{9}x = -\frac{2}{9}\]

Step 4: Find the term to complete the square

Identify the coefficient of \(x\), which is \(-\frac{13}{9}\), and calculate half of it squared:\[\left(\frac{-13}{18}\right)^2 = \frac{169}{324}\]Add this value to both sides of the equation:\[x^2 - \frac{13}{9}x + \frac{169}{324} = -\frac{2}{9} + \frac{169}{324}\]

Step 5: Simplify right side and express as a square

Simplify the right side:\[-\frac{2}{9} = -\frac{72}{324}\]\[-\frac{72}{324} + \frac{169}{324} = \frac{97}{324}\]Now the equation is:\[(x - \frac{13}{18})^2 = \frac{97}{324}\]

Step 6: Take the square root of both sides

Take the square root of both sides to solve for \(x\):\[x - \frac{13}{18} = \pm \sqrt{\frac{97}{324}}\]Calculate the square root:\[x - \frac{13}{18} = \pm \frac{\sqrt{97}}{18}\]

Step 7: Solve for x and round

Solve for \(x\) by adding \(\frac{13}{18}\) to both sides:\[x = \frac{13}{18} \pm \frac{\sqrt{97}}{18}\]Calculate the numerical solutions and round to the nearest hundredth:\[x_1 = \frac{13}{18} + \frac{\sqrt{97}}{18} \approx 1.69\]\[x_2 = \frac{13}{18} - \frac{\sqrt{97}}{18} \approx -0.13\]

Key Concepts

Quadratic Equations

Quadratic equations are equations of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). These equations describe parabolas when graphed in the coordinate plane. The solutions to a quadratic equation are where the graph crosses the x-axis.
These points can be found using different methods:

• Factoring, if the equation can be expressed as a product of binomials.
• Using the quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), which applies to any quadratic equation.
• Completing the square, a method that involves forming a perfect square trinomial.

Each method has its own merits, with completing the square offering a neat algebraic technique to understand transformations of quadratic functions better.

Solving Equations

Solving equations involves finding the value of the variable that makes the equation true. In the context of quadratic equations, solving often means isolating \( x \) or finding its possible values.


Completing the square is a common technique to solve quadratic equations. It transforms the quadratic into a simpler expression in the form \( (x - p)^2 = q \). Here's how to do it:

• Ensure the quadratic term's coefficient is 1.
• Identify the linear coefficient and divide it by 2, then square the result.
• Add and subtract this square within the equation.
• The left side forms a perfect square trinomial, allowing you to factor it as \((x-n)^2\).
• Take the square root of both sides to solve for \( x \).

This method is particularly useful when equations don't factor easily or when graphing is the next logical step.

Algebraic Manipulation

Algebraic manipulation is a fundamental aspect of solving equations. It involves rearranging and simplifying expressions to isolate variables or simplify terms.

In solving quadratic equations by completing the square, algebraic manipulation includes several key steps:

• Expanding expressions by multiplying terms using the distributive property, like \( 9x(x-1) - 2(2x-1) \).
• Combining like terms, such as adding or subtracting similar variables or constants.
• Isolating the quadratic and linear terms from the constant to complete the square.
• Dividing to make the leading coefficient of the quadratic term 1, facilitating easier perfect square formation.

Each manipulation must maintain the equation's balance, ensuring that each side remains equal, preserving the equation's integrity and leading to correct solutions.