Discovering the Roots: Simplify x² - 2bx + ax - 2ba + 7x - 14b Using Quadratic Formula

Q: How do I know which terms to group together?

Look for terms that share common factors. Group $ x^2 + (a+7)x $ because both have x, and group $ -2bx - 2ba - 14b $ because all have factor -2b.

Q: What if I can't find a common factor in my groups?

Try rearranging the original terms differently! Sometimes you need to reorder before grouping. The goal is making each group have an obvious common factor.

Q: How do I verify my factored form is correct?

Expand your answer using FOIL or distribution. If you get back to $ x^2 - 2bx + ax - 2ba + 7x - 14b $, you factored correctly!

Polynomial Factorization with Multiple Variables

Use the root formula and extract the trinomial

$x^2-2bx+ax-2ba+7x-14b$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

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 written solution

Follow each step carefully to understand the complete solution

Understand the problem

Use the root formula and extract the trinomial

$x^2-2bx+ax-2ba+7x-14b$

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 $(x+7+a)(x-2b)$ .

Final Answer

$(x+7+a)(x-2b)$

Key Points to Remember

Essential concepts to master this topic

Grouping Rule: Rearrange terms to form pairs with common factors
Technique: Factor x² + (a+7)x becomes x(x+a+7) first
Check: Expand (x+7+a)(x-2b) to verify original expression ✓

Common Mistakes

Avoid these frequent errors

Factoring terms randomly without systematic grouping
Don't just factor individual terms like x² and -2bx separately = incomplete factorization! This leaves middle terms unfactored and prevents finding the final answer. Always group terms strategically so each group shares a common factor.

Practice Quiz

Test your knowledge with interactive questions

a = coefficient of x²

b = coefficient of x

c = coefficient of the constant term

What is the value of $$ c $$ in the function $$ y=-x^2+25x $$ ?

FAQ

Everything you need to know about this question

Why can't I just use the quadratic formula here?

The quadratic formula works for equations set equal to zero, but this problem asks you to factor an expression. Factoring by grouping is the right method when you have multiple variables like a and b.

How do I know which terms to group together?

Look for terms that share common factors. Group $x^2 + (a+7)x$ because both have x, and group $-2bx - 2ba - 14b$ because all have factor -2b.

What if I can't find a common factor in my groups?

Try rearranging the original terms differently! Sometimes you need to reorder before grouping. The goal is making each group have an obvious common factor.

How do I verify my factored form is correct?

Expand your answer using FOIL or distribution. If you get back to $x^2 - 2bx + ax - 2ba + 7x - 14b$ , you factored correctly!

Can this expression be factored in different ways?

The final factored form is unique, but you might group terms differently during the process. As long as your method is systematic and you factor completely, you'll get $(x+7+a)(x-2b)$ .

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