Algebraic Expression: Simplify And Evaluate

Let's dive into the world of algebraic expressions and tackle this problem together! We're going to simplify the expression $3x(4x - 5) + 3$ and then find its value for two different inputs: $x=3$ and x=rac{1}{2}. This is a fantastic way to practice your algebra skills and see how a single expression can represent different numbers depending on the value of the variable.

Simplifying the Algebraic Expression

Our journey begins with simplifying the expression $3x(4x - 5) + 3$. The first step in simplifying algebraic expressions is often to distribute any terms that are multiplied by a parenthetical expression. In this case, we have $3x$ multiplying the $(4x - 5)$ inside the parentheses. So, we multiply $3x$ by $4x$ and then $3x$ by $-5$.

• Multiplying $3x$ by $4x$: When we multiply terms with variables, we multiply their coefficients (the numbers in front) and add their exponents. So, $3 * 4 = 12$, and $x * x = x^{1+1} = x^2$. Putting it together, $3x * 4x = 12x^2$.
• Multiplying $3x$ by $-5$: Here, we multiply the coefficient $3$ by $-5$, which gives us $-15$. The variable $x$ remains as it is. So, $3x * -5 = -15x$.

Now, let's substitute these results back into our original expression: $12x^2 - 15x + 3$. At this point, we check if there are any like terms that can be combined. Like terms are terms that have the exact same variable raised to the exact same power. In $12x^2 - 15x + 3$, we have an $x^2$ term, an $x$ term, and a constant term. None of these are like terms, so our simplified expression is $12x^2 - 15x + 3$. This is the most basic form of our original expression.

Evaluating the Expression for $x=3$

Now that we have our simplified expression, $12x^2 - 15x + 3$, we can find its value when $x=3$. This means we will substitute every instance of $x$ in the simplified expression with the number 3. It's important to use parentheses when substituting, especially when dealing with exponents or negative numbers, to avoid errors. So, we replace $x$ with $(3)$:

$12(3)^2 - 15(3) + 3$

Following the order of operations (PEMDAS/BODMAS), we first handle the exponent. $3^2$ means $3 * 3$, which equals 9.

Now, our expression looks like this: $12(9) - 15(3) + 3$.

Next, we perform the multiplications:

• $12 * 9$: $12 * 9 = 108$.
• $15 * 3$: $15 * 3 = 45$.

Our expression now becomes: $108 - 45 + 3$.

Finally, we perform the addition and subtraction from left to right:

• $108 - 45$: $108 - 45 = 63$.
• $63 + 3$: $63 + 3 = 66$.

So, the value of the expression $3x(4x - 5) + 3$ when $x=3$ is $66$. Pretty straightforward, right?

Evaluating the Expression for x=rac{1}{2}

Our next task is to find the value of the same simplified expression, $12x^2 - 15x + 3$, but this time when x = rac{1}{2}. Again, we substitute every $x$ with (rac{1}{2}). Be prepared for a few fractions here – it's all part of the fun of algebra!

12(rac{1}{2})^2 - 15(rac{1}{2}) + 3

Let's start with the exponent: (rac{1}{2})^2. This means rac{1}{2} * rac{1}{2}. To multiply fractions, you multiply the numerators together and the denominators together. So, $1 * 1 = 1$ and $2 * 2 = 4$. Therefore, (rac{1}{2})^2 = rac{1}{4}.

Our expression now looks like this: 12(rac{1}{4}) - 15(rac{1}{2}) + 3.

Next, we perform the multiplications:

• 12 * rac{1}{4}: This is the same as rac{12}{1} * rac{1}{4}. Multiplying across, we get rac{12 * 1}{1 * 4} = rac{12}{4}. This fraction simplifies to $3$.
• 15 * rac{1}{2}: Similar to the above, this is rac{15}{1} * rac{1}{2} = rac{15 * 1}{1 * 2} = rac{15}{2}. This fraction cannot be simplified further into a whole number, so we leave it as rac{15}{2}.

Now, our expression is: 3 - rac{15}{2} + 3.

We can combine the whole numbers first: $3 + 3 = 6$.

So, we have: 6 - rac{15}{2}.

To subtract a fraction from a whole number, we need to express the whole number as a fraction with the same denominator as the fraction we are subtracting. Our denominator is 2, so we need to write 6 as a fraction with a denominator of 2. We know that 6 = rac{6}{1}. To get a denominator of 2, we multiply both the numerator and the denominator by 2: rac{6 * 2}{1 * 2} = rac{12}{2}.

Now, our expression is: rac{12}{2} - rac{15}{2}.

Since the denominators are the same, we can subtract the numerators: $12 - 15 = -3$. The denominator stays the same.

So, the final result is rac{-3}{2}. This can also be written as -rac{3}{2} or as a mixed number, -1rac{1}{2}, or as a decimal, $-1.5$.

Conclusion

We've successfully simplified the algebraic expression $3x(4x - 5) + 3$ to $12x^2 - 15x + 3$. Then, we evaluated this simplified expression for two different values of $x$. When $x=3$, the expression equals $66$, and when x=rac{1}{2}, the expression equals -rac{3}{2}. This demonstrates the power of algebraic expressions to represent a rule that can be applied to various numbers. Understanding how to simplify and evaluate expressions is a foundational skill in mathematics, opening doors to more complex problem-solving.

If you'd like to explore more about algebraic simplification and evaluation, you can check out resources from Khan Academy or Math is Fun.