Which model represent the factors of 4x^2-9
The correct answer and explanation is:
The expression given is:
4x² – 9
This is a classic example of a difference of squares. The difference of squares formula is:
a2−b2=(a−b)(a+b)a^2 – b^2 = (a – b)(a + b)
Step 1: Identify the squares
- 4x24x^2 is a perfect square because it can be written as (2x)2(2x)^2.
- 99 is a perfect square because it can be written as 323^2.
Step 2: Apply the difference of squares formula
Using a=2xa = 2x and b=3b = 3, the expression factors as: 4×2−9=(2x)2−32=(2x−3)(2x+3)4x^2 – 9 = (2x)^2 – 3^2 = (2x – 3)(2x + 3)
Final factorization:
4×2−9=(2x−3)(2x+3)4x^2 – 9 = (2x – 3)(2x + 3)
Explanation:
Factoring expressions is a process of rewriting a polynomial as a product of simpler polynomials. One common type of factoring involves recognizing patterns such as the difference of squares. The difference of squares occurs when you have two perfect squares being subtracted, as in this problem.
Here, the polynomial 4×2−94x^2 – 9 looks like a2−b2a^2 – b^2, where both 4x24x^2 and 99 are squares:
- 4×2=(2x)24x^2 = (2x)^2 since 2x×2x=4x22x \times 2x = 4x^2.
- 9=329 = 3^2.
The formula for the difference of squares states that such an expression can be factored into the product of two binomials: one with the sum of aa and bb, and one with their difference.
Therefore, 4×2−94x^2 – 9 factors into (2x−3)(2x+3)(2x – 3)(2x + 3).
This factorization is useful in simplifying expressions, solving quadratic equations, and finding roots. For instance, if you set the expression equal to zero: 4×2−9=0 ⟹ (2x−3)(2x+3)=04x^2 – 9 = 0 \implies (2x – 3)(2x + 3) = 0
Using the zero product property: 2x−3=0or2x+3=02x – 3 = 0 \quad \text{or} \quad 2x + 3 = 0
Solving these gives the roots: x=32orx=−32x = \frac{3}{2} \quad \text{or} \quad x = -\frac{3}{2}
This shows how factoring helps find solutions to polynomial equations