Simplify x^2+1/2(20x^2-10x+6)+2x
The problem presents an algebraic expression and asks to simplify it. Simplifying an algebraic expression typically involves combining like terms, distributing multiplication over addition or subtraction (applying the distributive property), and performing arithmetic operations to obtain the simplest form of the expression. In this case, it will involve distributing the 1/2 across the trinomial (20x^2-10x+6), combining the resulting like terms with x^2 and 2x, and then condensing the expression to its most reduced form.
$x^{2} + \frac{1}{2} \left(\right. 20 x^{2} - 10 x + 6 \left.\right) + 2 x$
Answer
Solution:
Step 1: Break down each term for simplification.
Step 1.1: Utilize the distributive property to expand the expression.
$x^2 + \frac{1}{2}(20x^2) + \frac{1}{2}(-10x) + \frac{1}{2} \cdot 6 + 2x$
Step 1.2: Proceed to simplify the terms.
Step 1.2.1: Identify and remove common factors.
Step 1.2.1.1: Extract the factor of 2 from $20x^2$.
$x^2 + \frac{1}{2}(2(10x^2)) + \frac{1}{2}(-10x) + \frac{1}{2} \cdot 6 + 2x$
Step 1.2.1.2: Eliminate the common factor.
$x^2 + \frac{1}{\cancel{2}}(\cancel{2}(10x^2)) + \frac{1}{2}(-10x) + \frac{1}{2} \cdot 6 + 2x$
Step 1.2.1.3: Reconstruct the expression.
$x^2 + 10x^2 + \frac{1}{2}(-10x) + \frac{1}{2} \cdot 6 + 2x$
Step 1.2.2: Again, remove any common factors.
Step 1.2.2.1: Factor out 2 from $-10x$.
$x^2 + 10x^2 + \frac{1}{2}(2(-5x)) + \frac{1}{2} \cdot 6 + 2x$
Step 1.2.2.2: Cancel out the common factor.
$x^2 + 10x^2 + \frac{1}{\cancel{2}}(\cancel{2}(-5x)) + \frac{1}{2} \cdot 6 + 2x$
Step 1.2.2.3: Rewrite the simplified terms.
$x^2 + 10x^2 - 5x + \frac{1}{2} \cdot 6 + 2x$
Step 1.2.3: Remove common factors one last time.
Step 1.2.3.1: Extract the factor of 2 from 6.
$x^2 + 10x^2 - 5x + \frac{1}{2} \cdot (2 \cdot 3) + 2x$
Step 1.2.3.2: Eliminate the common factor.
$x^2 + 10x^2 - 5x + \frac{1}{\cancel{2}} \cdot (\cancel{2} \cdot 3) + 2x$
Step 1.2.3.3: Finalize the expression.
$x^2 + 10x^2 - 5x + 3 + 2x$
Step 2: Combine like terms to simplify the expression.
Step 2.1: Sum the terms $x^2$ and $10x^2$.
$11x^2 - 5x + 3 + 2x$
Step 2.2: Add together $-5x$ and $2x$.
$11x^2 - 3x + 3$
Knowledge Notes:
To simplify an algebraic expression, one must perform several steps, including:
Distributive Property: This property allows us to multiply a single term by each term within a parenthesis. It is expressed as $a(b+c) = ab + ac$.
Combining Like Terms: This involves adding or subtracting terms that have the same variable raised to the same power. For example, $ax^n + bx^n = (a+b)x^n$.
Factoring: This is the process of finding numbers or expressions that multiply together to give the original number or expression. For instance, factoring 2 out of 4 gives $2 \times 2$.
Simplification: This process involves reducing an expression to its simplest form by performing all possible operations, including addition, subtraction, multiplication, division, and factoring out common factors.
In the given solution, the expression is simplified by applying the distributive property, factoring out common factors, and combining like terms. The use of Latex in the solution helps to clearly display mathematical expressions and operations.
