Question

Factor \(3 x^2-8 x+4\) A. \((x-4)(3 x-1)\) B. \((x-1)(3 x-4)\) C. \((3 x-1)(3 x-4)\) D. \((x-2)(3 x-2)\)

Short answer

The factorization of \( 3 x^2 - 8 x + 4 \) is \( (x - 2)(3 x - 2) \). Thus, the correct answer is \( D \).

Step-by-step solution

- Identify the quadratic coefficients

For the quadratic expression in the form \( a x^2 + b x + c \), identify the coefficients: \( a = 3 \), \( b = -8 \), and \( c = 4 \).

- Find the product of a and c

Multiply the coefficients \( a \) and \( c \): \( 3 \times 4 = 12 \).

- Find two numbers that multiply to ac and add to b

We need to find two numbers that multiply to 12 and add to -8. These numbers are -6 and -2, because \( -6 \times -2 = 12 \) and \( -6 + -2 = -8. \)

- Rewrite the middle term

Break the middle term \( -8x \) into two terms using the numbers found: \( 3 x^2 - 6 x - 2 x + 4 \).

- Factor by grouping

Group the terms to factor by grouping: \( 3 x (x - 2) - 2 (x - 2) \).

- Factor out the common binomial

Factor out the common binomial \( (x - 2) \): \( (3 x - 2)(x - 2) \).

Key concepts

quadratic coefficients

Quadratic equations are generally written in the form of ax^2 + bx + ca, where 'a', 'b', and 'c' are called the quadratic coefficients. These coefficients determine the shape and position of the parabola graph represented by the quadratic equation.
In our example, the equation is 3x^2 - 8x + 4. Here, the coefficients are:

• 'a' = 3
• 'b' = -8
• 'c' = 4

Remember, 'a' is the coefficient of x^2, 'b' is the coefficient of x, and 'c' is the constant term.
Understanding these coefficients helps us decide the best method to factorize the quadratic equation.

factoring by grouping

Factoring by grouping involves breaking the middle term of a quadratic equation so that you can factorize the expression into simpler terms. After identifying the quadratic coefficients, the next step is to find two numbers that multiply to a * c and add up to b.For example, in 3x^2 - 8x + 4, 'a' = 3 and 'c' = 4, we first calculate a * c:

3 * 4 = 12.
Next, we need two numbers that multiply to 12 and add up to -8. These numbers are -6 and -2, because -6 * -2 = 12 and -6 + -2 = -8.

Next, we replace the middle term with these two numbers: 3x^2 - 6x - 2x + 4.
Now we group and factor: 3x(x - 2) - 2(x - 2).
Notice the common binomial, and factor it out: (3x - 2)(x - 2).
This process shows how to break down and factorize a quadratic equation using the grouping method.

product-sum method

The product-sum method is a useful technique for factoring quadratic equations when the leading coefficient is not one. It involves finding two numbers whose product equals a * c and whose sum equals b.

Taking our given quadratic equation 3x^2 - 8x + 4, with coefficients 'a' = 3, 'b' = -8, and 'c' = 4, first compute a * c, which equals 12 (since 3 * 4 = 12). The next step is finding two numbers that multiply to 12 and add up to -8. Those numbers are -6 and -2.

• -6 * -2 = 12
• -6 + -2 = -8

Using these numbers, rewrite the middle term -8x as -6x - 2x, so the expression becomes 3x^2 - 6x - 2x + 4.
Then, use grouping to factorize: group the terms as 3x(x - 2) - 2(x - 2) and then factor out the common binomial (x - 2), which results in (3x - 2)(x - 2). This is how the product-sum method simplifies factoring.