Factorize 12x2 - 24x + 9. 2. Find C such that x2 - 2x + C is a perfect square trinomial.

Here are the solutions to the problems. 1. Factorize 12x^2 - 24x + 9 Step 1: Identify coefficients and find the product ac and sum b. For a quadratic expression ax^2 + bx + c, we have a=12, b=-24, c=9. The product ac = 12 × 9 = 108. The sum b = -24. Step 2: Find two numbers that multiply to 108 and add to -24. The two numbers are -18 and -6. (-18) × (-6) = 108 (-18) + (-6) = -24 Step 3: Rewrite the middle term using these two numbers. 12x^2 - 24x + 9 = 12x^2 - 18x - 6x + 9 Step 4: Factor by grouping. Group the first two terms and the last two terms: (12x^2 - 18x) + (-6x + 9) Factor out the greatest common factor from each group: 6x(2x - 3) - 3(2x - 3) Step 5: Factor out the common binomial. (6x - 3)(2x - 3) We can factor out a common factor of 3 from the first binomial: 3(2x - 1)(2x - 3) The fully factored form is 3(2x - 1)(2x - 3). 2. Find C such that x^2 - 2x + C is a perfect square trinomial, and write the trinomial. Step 1: Recall the condition for a perfect square trinomial. For a quadratic expression ax^2 + bx + c to be a perfect square trinomial, c = ((b)/(2a))^2. For x^2 - 2x + C, we have a=1 and b=-2. Step 2: Calculate the value of C. C = ((-2)/(2 × 1))^2 = ((-2)/(2))^2 = (-1)^2 = 1 Step 3: Write the perfect square trinomial and its factored form. The perfect square trinomial is x^2 - 2x + 1. This can be factored as (x - 1)^2. The value of C is 1, and the trinomial is x^2 - 2x + 1. 3. Find C such that x^2 + 8x + C is a perfect square trinomial, and write the trinomial. Step 1: Identify coefficients. For x^2 + 8x + C, we have a=1 and b=8. Step 2: Calculate the value of C. C = ((8)/(2 × 1))^2 = ((8)/(2))^2 = (4)^2 = 16 Step 3: Write the perfect square trinomial and its factored form. The perfect square trinomial is x^2 + 8x + 16. This can be factored as (x + 4)^2. The value of C is 16, and the trinomial is x^2 + 8x + 16. Send me the next one 📸