Problem 11 - Elementary Algebra

Question

Factor. $$ 9 x 2-12 x+4 $$

Step-by-Step Solution

Answer

The expression factors to \((3x - 2)^2\).

Step 1: Identify A, B, and C

For the quadratic expression, rewrite it in the standard form, ensuring it is in the format \(ax^2 + bx + c\). Here, the expression is already given as \(9x^2 - 12x + 4\). Identify \(A = 9\), \(B = -12\), and \(C = 4\).

Step 2: Calculate the Discriminant

Calculate the discriminant using the formula \(B^2 - 4AC\). Substitute the identified values: \((-12)^2 - 4 \times 9 \times 4 = 144 - 144 = 0\).

Step 3: Understand the Implication of the Discriminant

The discriminant value is zero, indicating the quadratic is a perfect square and has one repeated solution.

Step 4: Write the Quadratic as a Perfect Square

Express the equation in the form \((px + q)^2\). Use this form: \( (3x - 2)^2\), as expanding this expression gives back \(9x^2 - 12x + 4\).

Step 5: Factored Form

The expression \((3x - 2)^2\) represents the factorization of the equation. Thus, the quadratic expression is factored into \( (3x - 2)(3x - 2)\).

Key Concepts

Discriminant Calculation

The discriminant is a valuable tool for understanding the nature of a quadratic equation. It's like a "magic number" that tells us how many solutions our equation has, and what type they are. To find the discriminant, we use the formula: \[ B^2 - 4AC \] Here, \(B\), \(A\), and \(C\) are coefficients from the quadratic equation in the form \( ax^2 + bx + c \). The discriminant will help us decide the next steps in solving the quadratic.
In the given example, we have:

• \(A = 9\)
• \(B = -12\)
• \(C = 4\)

Substituting these into the discriminant formula, we calculate: \[ (-12)^2 - 4 imes 9 imes 4 = 144 - 144 = 0 \] This result is especially interesting because a discriminant of 0 implies there is exactly one solution, a special situation where the quadratic is a perfect square.

Perfect Square

When a quadratic has a discriminant of zero, it means the equation can be expressed as a "perfect square." This means that both roots or solutions of the equation are identical, and so it can be written in a squared form.
To find this perfect square, we take the original equation and express it like this: \((px + q)^2\). This represents taking the same factor and multiplying it by itself.
Consider the quadratic from the example:

• \(9x^2 - 12x + 4\)

Here, the perfect square form is \((3x - 2)^2\). This is derived because expanding \((3x - 2)(3x - 2)\) gives us back to \(9x^2 - 12x + 4\). Therefore, the quadratic can be factored into a neat square form, reflecting its single solution.

Factored Form

The factored form of a quadratic is typically expressed when a quadratic can be neatly represented as a product of its factors. This makes it easier to understand and solve the equation.
For quadratics where the discriminant is zero, as seen previously, the expression can be factored into a perfect square. This form will look like \((px+q)(px+q)\), essentially \((px+q)^2\).
In our example with the quadratic \(9x^2 - 12x + 4\), we found that the factored form is \((3x - 2)^2\). This is powerful because it not only gives the factorization but also indicates the solutions (or roots) of the quadratic. Since it is a perfect square, it implies the root \(3x - 2 = 0\), leading to the solution \(x = \frac{2}{3}\). Thus, recognizing factored form is crucial in revealing valuable information about the quadratic's behavior and solutions.