Problem 54 - Algebra 2

Question

Solve each equation by completing the square. \(3 x^{2}-4 x=2\)

Step-by-Step Solution

Answer

The solutions are \(x = \frac{2 + \sqrt{10}}{3}\) and \(x = \frac{2 - \sqrt{10}}{3}\).

Step 1: Move constant to the other side

Start by isolating the quadratic and linear terms of the quadratic equation. Rearrange the equation by moving the constant term to the right side:\[ 3x^2 - 4x - 2 = 0 \] becomes \[ 3x^2 - 4x = 2 \]

Step 2: Divide by the coefficient of \(x^2\)

Divide every term by the leading coefficient (which is 3 in this case) to make the coefficient of \(x^2\) equal to 1.\[ x^2 - \frac{4}{3}x = \frac{2}{3} \]

Step 3: Complete the square

To complete the square, take half of the coefficient of \(x\) (\(-\frac{4}{3}\)), square it, and add it to both sides of the equation.Half of \(-\frac{4}{3}\) is \(-\frac{2}{3}\), and its square is \(\frac{4}{9}\).Add \(\frac{4}{9}\) to both sides:\[ x^2 - \frac{4}{3}x + \frac{4}{9} = \frac{2}{3} + \frac{4}{9} \]

Step 4: Write the left side as a square

The left side is now a perfect square trinomial. Rewrite it as a squared binomial:\[ \left(x - \frac{2}{3}\right)^2 = \frac{2}{3} + \frac{4}{9} \]

Step 5: Simplify the right side

Simplify the right side by finding a common denominator and adding the fractions:\[ \frac{2}{3} = \frac{6}{9} \]Thus:\[ \frac{6}{9} + \frac{4}{9} = \frac{10}{9} \]The equation becomes:\[ \left(x - \frac{2}{3}\right)^2 = \frac{10}{9} \]

Step 6: Solve for \(x\)

Take the square root of both sides to solve for \(x\). Remember to consider both the positive and negative roots:\[ x - \frac{2}{3} = \pm\sqrt{\frac{10}{9}} \]Simplify:\[ x - \frac{2}{3} = \pm \frac{\sqrt{10}}{3} \]

Step 7: Isolate \(x\)

Add \(\frac{2}{3}\) to both sides to isolate \(x\):\[ x = \frac{2}{3} \pm \frac{\sqrt{10}}{3} \]So the solutions are:\[ x = \frac{2 + \sqrt{10}}{3} \] and \[ x = \frac{2 - \sqrt{10}}{3} \]

Key Concepts

Quadratic Equation

A quadratic equation is one of the most common forms of polynomial equations. It’s generally written as:

• \( ax^2 + bx + c = 0 \)

Here:

• \( a \), \( b \), and \( c \) are constants, with \( a eq 0 \).
• The highest exponent of the variable \( x \) is 2, which makes it "quadratic."

Completing the square is one method to solve quadratic equations, which involves converting the equation into a perfect square trinomial. Once you have a trinomial, it can easily be rewritten as a square of a binomial, then solved using square roots. This method is particularly useful for equations where factoring is not straightforward, as it enables us to neatly isolate \( x \). Understanding quadratic equations is foundational for tackling more complex algebraic problems.

Perfect Square Trinomial

A perfect square trinomial is an expression that results from squaring a binomial. It typically takes the form:

• \( (x + p)^2 = x^2 + 2px + p^2 \)

For the given equation, completing the square meant transforming \( x^2 - \frac{4}{3}x \) into a perfect square trinomial. When you perform this operation, you’re focusing on finding a constant \( p^2 \) that turns the quadratic expression into a perfect square:

• Add half of the coefficient of \( x \), squared, to both sides of the equation.
• For example, take \(-\frac{4}{3} \), halve it to get \(-\frac{2}{3} \), and square to find \( \frac{4}{9} \).

Thus, the left side becomes \( \left(x - \frac{2}{3}\right)^2 \), a neat squared binomial. Recognizing and forming perfect square trinomials is key to simplifying and solving quadratic equations through the method of completing the square.

Binomial

A binomial is a polynomial with exactly two terms. In the context of solving quadratic equations by completing the square, the process often involves rewriting a quadratic expression as a square of a binomial.In our example, the expression \( (x - \frac{2}{3})^2 \) is a binomial squared. This is achieved by adjusting the quadratic expression into a perfect square trinomial that resembles \( (x - p)^2 \). The key benefits of structuring an equation as a squared binomial include:

• Simplified solving process through taking square roots.
• Clear expression of solutions when \( x \) is isolated more efficiently.

Understanding how to identify and utilize binomials is essential in algebra and can make solving equations cleaner and faster.

Square Root

The square root function is crucial when solving equations that have been restructured into squares of binomials. In essence, to solve an equation like \( (x - \frac{2}{3})^2 = \frac{10}{9} \), the next step is to apply the square root to both sides to "remove" the square, resulting in:

• \( x - \frac{2}{3} = \pm \frac{\sqrt{10}}{3} \)

Remember:

• The square root symbol \( \sqrt{ } \) signifies finding a number which, when multiplied by itself, equals the given value.
• Always consider both the positive and negative roots when solving for \( x \), since both will satisfy \( (x - \frac{2}{3})^2 = \frac{10}{9} \).

By utilizing the square root method, solutions often simplify into two possible values — one positive, one negative — neatly giving the set of solutions necessary for quadratic equations. This makes understanding square roots an invaluable skill in mathematics.