Can someone please show me step by step how to do these. I've read the book and there examples and It is just not clicking.

Directions: Factor each of the following polynomials completely

Problem:#1

14x^2-20x+6

Directions:Find all positive values for "K" for which each of the following can be factored.

Problem# 2

x^2 + x - K

I'll appreciate it...

14x^2-20x+6 = 2 (7x^2 -10x + 3)

You then attempt to factor
7x^2 -10x + 3 as follows:

7x^2 -10x + 3 = (ax + b)(cx + d)

Then a*c = 7 and b*d = 3 This then leads to:

7x^2 -10x + 3 = (7x + b)(x + d)

The coefficient of x is -10, if you work out the product you see that it is:

7*d + b

This means that you can take d = -1 and b = -3:

7x^2 -10x + 3 = (7x - 3)(x - 1)

thank you very much.

What is the value of x in the equation x 0.97 = −2?(1 point)

You're welcome! I'm glad I could help. Let me explain the steps in detail to factor the polynomial equation 14x^2 - 20x + 6:

Step 1: Look for the greatest common factor (GCF) of all the terms. In this case, the GCF is 2. You can factor out 2 from each term:
2(7x^2 - 10x + 3)

Step 2: Now we need to factor the remaining trinomial expression, which is 7x^2 - 10x + 3. We need to find two binomials that, when multiplied, will give us this trinomial.

Step 3: Set up the factored form as (ax + b)(cx + d). We need to find values for a, b, c, and d.

Step 4: The product of a and c should be equal to the coefficient of x^2, which is 7 in this case. Since 7 is a prime number, the only possible values for a and c are 1 and 7.

Step 5: The product of b and d should be equal to the constant term, which is 3. In this case, b and d can be 1 and 3, respectively.

Step 6: Now, rewrite the trinomial expression using the values of a, b, c, and d that we found:
7x^2 - 10x + 3 = (7x + 1)(x + 3)

So, the completely factored form of the polynomial 14x^2 - 20x + 6 is:
14x^2 - 20x + 6 = 2(7x + 1)(x + 3)

Now let's move on to the second problem:

To find all positive values for "K" for which the polynomial x^2 + x - K can be factored, we need to determine what values of K will make the quadratic equation factorable.

For a quadratic equation to be factorable, its discriminant (b^2 - 4ac) must be greater than or equal to 0.

In this case, the quadratic equation is x^2 + x - K, where a = 1, b = 1, and c = -K.

The discriminant is given by D = b^2 - 4ac.

Replacing the respective values, we have D = (1)^2 - 4(1)(-K) = 1 + 4K.

For the equation to be factorable, D must be greater than or equal to 0. Therefore, 1 + 4K ≥ 0.

Simplifying the inequality, we get 4K ≥ -1, which can be further simplified to K ≥ -1/4.

So, all positive values of K that satisfy the condition and make the polynomial x^2 + x - K factorable are K ≥ -1/4.

I hope this explanation helps you understand the steps involved in factoring polynomials and finding values that make quadratic equations factorable. If you have any further questions, feel free to ask!