Problem 14 - Algebra and Trigonometry

Question

Solve the equation by factoring. $$ 3 x^{2}+1=4 x $$

Step-by-Step Solution

Answer

The solutions are \( x = \frac{1}{3} \) and \( x = 1 \).

Step 1: Rearrange the Equation

To factor the equation, start by rearranging it into a standard quadratic form. Subtract \( 4x \) from both sides to set the equation to equal zero: \[ 3x^2 + 1 - 4x = 0 \] Reorder the terms: \[ 3x^2 - 4x + 1 = 0 \]

Step 2: Identify a, b, and c

In the quadratic equation \( ax^2 + bx + c = 0 \), identify the coefficients:- \( a = 3 \)- \( b = -4 \)- \( c = 1 \)

Step 3: Factor the Quadratic

Look for two numbers that multiply to \( a \cdot c = 3 \times 1 = 3 \) and add to \( b = -4 \). These numbers are \(-3\) and \(-1\). Use these to break the middle term and factor by grouping:\[ 3x^2 - 3x - x + 1 = 0 \]Factor the first two and last two terms separately:\[ 3x(x - 1) - 1(x - 1) = 0 \]Factor out the common term \((x - 1)\):\[ (3x - 1)(x - 1) = 0 \]

Step 4: Solve for x

Set each factor equal to zero and solve for \( x \):- For \( 3x - 1 = 0 \):\[ 3x = 1 \]\[ x = \frac{1}{3} \]- For \( x - 1 = 0 \):\[ x = 1 \]

Key Concepts

Factoring Quadratics

Factoring quadratics is a method used to simplify and solve quadratic equations by expressing them as the product of two binomials. It is often used when the coefficients are manageable and allow for easy decomposition. The key is to rewrite the equation in the standard form, \[ ax^2 + bx + c = 0 \]
where you can identify the coefficients:

• \( a \) - the coefficient of \( x^2 \)
• \( b \) - the coefficient of \( x \)
• \( c \) - the constant term.

The next step is to find two numbers that multiply to the product of the coefficient \( a \) and the constant \( c \) and add up to the coefficient \( b \).
Once these numbers are found, split the middle term using them. After splitting, you can factor by grouping, which involves looking for pairs of terms that have common factors. This allows you to factor out binomials, leading to an easier path to solve the equation.

Solving Equations

Solving equations involves finding the value of the unknown variable that makes the equation true. For quadratic equations, like \( (3x - 1)(x - 1) = 0 \), solving involves using the Zero Product Property. This property states that if a product of two factors equals zero, at least one of the factors must be zero.
To solve the factored form of a quadratic equation:

• Set each factor equal to zero separately.
• For example, \( 3x - 1 = 0 \) and \( x - 1 = 0 \).
• Solve each of these simple linear equations to find the possible values for \( x \).

Through this method, the solutions \( x = \frac{1}{3} \) and \( x = 1 \) are identified as the points where the original quadratic equation equals zero.

Quadratic Formula

The quadratic formula is a versatile tool that can solve any quadratic equation. It's especially useful when factoring is not easily applicable. The formula is:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here's how it works:

• Start by ensuring the equation is in standard form \( ax^2 + bx + c = 0 \).
• Identify the coefficients \( a \), \( b \), and \( c \).
• Plug these values into the formula.
• Calculate the discriminant \( b^2 - 4ac \), which determines the nature of the roots.

Using the quadratic formula offers a robust method to find solutions even when factoring is challenging.