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Math  /  Algebra

QuestionFactor 2x2−7x+62 x^{2}-7 x+6 a. (x−3)(2x−2)(x-3)(2 x-2) c. (x−2)(2x−3)(x-2)(2 x-3) b. (x−2)(2x+3)(x-2)(2 x+3) d. (x−2)(x−3)(x-2)(x-3)

Studdy Solution

STEP 1

1. The quadratic expression 2x2−7x+62x^2 - 7x + 6 can be factored into two binomials of the form (ax+b)(cx+d)(ax + b)(cx + d).
2. The coefficients of the binomials must satisfy the conditions derived from expanding (ax+b)(cx+d)(ax + b)(cx + d).

STEP 2

1. Identify the potential forms of factors based on the given options.
2. Expand each given option to see if it matches the original quadratic expression.
3. Confirm the correct factorization by comparing the expanded form to the original expression.

STEP 3

Consider option a: (x−3)(2x−2)(x-3)(2x-2). Expand this product to see if it matches 2x2−7x+62x^2 - 7x + 6. (x−3)(2x−2)=x(2x−2)−3(2x−2) (x-3)(2x-2) = x(2x-2) - 3(2x-2) =2x2−2x−6x+6 = 2x^2 - 2x - 6x + 6 =2x2−8x+6 = 2x^2 - 8x + 6

STEP 4

The expansion of option a results in 2x2−8x+62x^2 - 8x + 6, which does not match 2x2−7x+62x^2 - 7x + 6. Thus, option a is incorrect.

STEP 5

Consider option b: (x−2)(2x+3)(x-2)(2x+3). Expand this product to see if it matches 2x2−7x+62x^2 - 7x + 6. (x−2)(2x+3)=x(2x+3)−2(2x+3) (x-2)(2x+3) = x(2x+3) - 2(2x+3) =2x2+3x−4x−6 = 2x^2 + 3x - 4x - 6 =2x2−x−6 = 2x^2 - x - 6

STEP 6

The expansion of option b results in 2x2−x−62x^2 - x - 6, which does not match 2x2−7x+62x^2 - 7x + 6. Thus, option b is incorrect.

STEP 7

Consider option c: (x−2)(2x−3)(x-2)(2x-3). Expand this product to see if it matches 2x2−7x+62x^2 - 7x + 6. (x−2)(2x−3)=x(2x−3)−2(2x−3) (x-2)(2x-3) = x(2x-3) - 2(2x-3) =2x2−3x−4x+6 = 2x^2 - 3x - 4x + 6 =2x2−7x+6 = 2x^2 - 7x + 6

STEP 8

The expansion of option c results in 2x2−7x+62x^2 - 7x + 6, which matches the original quadratic expression. Thus, option c is correct.

STEP 9

Consider option d: (x−2)(x−3)(x-2)(x-3). Expand this product to see if it matches 2x2−7x+62x^2 - 7x + 6. (x−2)(x−3)=x(x−3)−2(x−3) (x-2)(x-3) = x(x-3) - 2(x-3) =x2−3x−2x+6 = x^2 - 3x - 2x + 6 =x2−5x+6 = x^2 - 5x + 6

STEP 10

The expansion of option d results in x2−5x+6x^2 - 5x + 6, which does not match 2x2−7x+62x^2 - 7x + 6. Thus, option d is incorrect.
Solution: The correct factorization of 2x2−7x+62x^2 - 7x + 6 is option c: (x−2)(2x−3)(x-2)(2x-3).

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