Discovering the Roots: Simplify x^2 - 2bx + ax - 2ba + 7x - 14b Using Quadratic Formula

Video Solution

Solution Steps

00:00 Factor the trinomials using the root formula

00:03 Arrange the equation so we can identify the coefficients

00:24 Identify the coefficients

00:40 Use the root formula to find possible solutions

00:50 Substitute appropriate values and solve to find the solutions

01:19 Use the shortened multiplication formulas to open the parentheses

02:19 Collect terms

03:23 Factor back into trinomial form

03:46 Find the 2 possible solutions (addition and subtraction)

04:22 These are the solutions, with them we'll construct the trinomial

04:32 Find what makes each solution equal zero

04:43 These will be the multipliers of the trinomial

04:46 And this is the solution to the question

Step-by-Step Solution

To solve this problem, we must factor the expression $x^2 - 2bx + ax - 2ba + 7x - 14b$ . Let’s go through the steps involved:

Step 1: Rearrange the terms to enable easier grouping: $x^2 + (ax + 7x) + (-2bx - 2ba - 14b)$ .
Step 2: Group the terms in pairs that can be factored individually:
- First Group: $x^2 + (a+7)x$
- Second Group: $-2b(x + a) - 14b$
Step 3: Look for common factors for each group. Factor them out:
- First Group: Factor out $x$ , giving $x(x + a + 7)$
- Second Group: Factor out $-2b$ , giving $-2b(x + a + 7)$
Step 4: Factor by grouping common terms.
Both the expressions now have a common factor $(x + a + 7)$ , so we can express the original expression as: $(x - 2b)(x + a + 7)$ .
Step 5: Simplify and ensure correctness: Correctly reordered the terms within the parentheses as necessary to reflect the correct factorization.

Therefore, the expression $x^2 - 2bx + ax - 2ba + 7x - 14b$ can be factored as $(x+7+a)(x-2b)$ .