6x 7x - 5 - 7 5x

Simplifying algebraic expressions is a foundational skill in mathematics that bridges the gap between basic arithmetic and advanced calculus. Consider this: when faced with a string of terms like 6x 7x - 5 - 7 5x, the first step is always to clarify the operations implied by the spacing. In standard algebraic notation, terms written side-by-side usually imply multiplication, but in the context of "simplifying expressions" worksheets, missing plus signs often imply addition. This article provides a comprehensive, step-by-step guide to simplifying this expression under both standard interpretations, ensuring you master the core concepts of combining like terms and the order of operations And that's really what it comes down to. No workaround needed..

Understanding the Expression: Two Possible Interpretations

Before diving into the solution, we must define the expression precisely. Think about it: the input 6x 7x - 5 - 7 5x lacks explicit operators between the first two terms and the last two terms. In mathematics, ambiguity is the enemy of accuracy.

1. Scenario A (Addition/Subtraction Context): The missing operators are plus signs. The expression is $6x + 7x - 5 - 7 + 5x$. This is the most common format for "combining like terms" practice problems.
2. Scenario B (Multiplication Context): Juxtaposition implies multiplication. The expression is $(6x)(7x) - 5 - (7)(5x)$ or $42x^2 - 5 - 35x$. This appears in polynomial multiplication or distribution exercises.

We will solve both completely so you are prepared for either case.

Scenario A: Simplifying by Combining Like Terms ($6x + 7x - 5 - 7 + 5x$)

This scenario focuses on the fundamental skill of identifying and grouping like terms. Like terms are terms that have the exact same variable raised to the exact same power Most people skip this — try not to..

Step 1: Identify the Terms and Their Categories

Break the expression down into individual components separated by addition or subtraction signs:

• Variable Terms (x-terms): $6x$, $7x$, $5x$ (All have the variable $x$ to the power of 1).
• Constant Terms (Numbers): $-5$, $-7$ (Note: the subtraction signs belong to the numbers following them).

Step 2: Group Like Terms Using the Commutative Property

The Commutative Property of Addition allows us to rearrange terms without changing the value. Group the $x$-terms together and the constants together: $ (6x + 7x + 5x) + (-5 - 7) $

Step 3: Add the Coefficients of the Like Terms

• For the $x$-terms: Add the coefficients (the numbers in front of the variable). $ 6 + 7 + 5 = 18 $ Result: $18x$
• For the constants: Add the integers. $ -5 + (-7) = -12 $ Result: $-12$

Step 4: Write the Final Simplified Expression

Combine the results from Step 3: $ \mathbf{18x - 12} $

Step 5: Optional – Factoring (Greatest Common Factor)

Often, teachers ask for the expression in factored form. Identify the Greatest Common Factor (GCF) of 18 and 12.

• Factors of 18: 1, 2, 3, 6, 9, 18
• Factors of 12: 1, 2, 3, 4, 6, 12
• GCF = 6

Divide each term by 6: $ 6(3x - 2) $ Both $18x - 12$ and $6(3x - 2)$ are mathematically equivalent; the preferred form depends on the specific instructions (e.But g. , "simplify" vs. "factor completely") Surprisingly effective..

Scenario B: Simplifying with Multiplication ($6x \cdot 7x - 5 - 7 \cdot 5x$)

If the spaces imply multiplication, the expression contains different powers of $x$ ($x^2$ and $x$), meaning they are unlike terms and cannot be combined into a single term.

Step 1: Apply the Order of Operations (PEMDAS/BODMAS)

Multiplication comes before Subtraction. Perform the multiplication operations first.

1. First Term: $6x \cdot 7x$

   • Multiply coefficients: $6 \times 7 = 42$
   • Multiply variables: $x \times x = x^2$ (Add exponents: $1+1=2$)
   • Result: $42x^2$

2. Last Term: $7 \cdot 5x$

   • Multiply coefficients: $7 \times 5 = 35$
   • Variable remains: $x$
   • Result: $35x$

Step 2: Rewrite the Expression

Substitute the multiplied results back into the original structure: $ 42x^2 - 5 - 35x $

Step 3: Arrange in Standard Form

Standard form for polynomials orders terms by descending degree (highest exponent to lowest).

1. $42x^2$ (Degree 2)
2. $-35x$ (Degree 1)
3. $-5$ (Degree 0 / Constant)

Final Simplified Expression (Standard Form): $ \mathbf{42x^2 - 35x - 5} $

Step 4: Check for Further Simplification (Factoring)

Check if the trinomial $42x^2 - 35x - 5$ can be factored Most people skip this — try not to..

• Look for a GCF of all three terms: 42, 35, and 5 share no common factor other than 1.
• Check if it is a factorable quadratic (product $a \cdot c = 42 \cdot -5 = -210$, sum $b = -35$).
  • Factors of -210 that sum to -35? $-35 \times 6 = -210$, but $-35 + 6 = -29$. $15 \times -14 = -21

Continuing from the point where the quadratic (42x^{2}-35x-5) was examined for factorability, we see that no pair of integers multiplies to (-210) and adds to (-35). Because the discriminant

[ \Delta = (-35)^{2}-4\cdot 42\cdot (-5)=1225+840=2065 ]

is not a perfect square, the polynomial does not factor over the integers (or rationals). The only common factor among the three terms is 1, so the expression is already in its simplest factored form.

Solving the Equation (42x^{2}-35x-5=0)

Even though the quadratic cannot be factored nicely, it can be solved using the quadratic formula:

[ x=\frac{-b\pm\sqrt{\Delta}}{2a} =\frac{35\pm\sqrt{2065}}{84}. ]

Since (\sqrt{2065}\approx45.44), the two real roots are approximately

[ x_{1}\approx\frac{35+45.44}{84}\approx0.99,\qquad x_{2}\approx\frac{35-45.44}{84}\approx-0.11. ]

These values can be verified by substitution, confirming that the original expression indeed equals zero at the computed points.

Alternative Forms

If a teacher requests a “completed‑square” form, we can rewrite the quadratic as follows:

[ \begin{aligned} 42x^{2}-35x-5 &=42\left(x^{2}-\frac{35}{42}x\right)-5\[2pt] &=42\left[x^{2}-\frac{5}{6}x\right]-5\[2pt] &=42\left[\left(x-\frac{5

Continuing from the point where the completed‑square manipulation was begun:

[ \begin{aligned} 42x^{2}-35x-5 &=42\Bigl(x^{2}-\frac{35}{42}x\Bigr)-5\[2pt] &=42\Bigl(x^{2}-\frac{5}{6}x\Bigr)-5. \end{aligned} ]

To complete the square, take half of the linear coefficient (-\frac{5}{6}), which is (-\frac{5}{12}), and square it:

[ \left(\frac{5}{12}\right)^{2}=\frac{25}{144}. ]

Add and subtract this quantity inside the brackets:

[ \begin{aligned} 42\Bigl[\left(x-\frac{5}{12}\right)^{2}-\frac{25}{144}\Bigr]-5 &=42\left(x-\frac{5}{12}\right)^{2}-42\cdot\frac{25}{144}-5\[2pt] &=42\left(x-\frac{5}{12}\right)^{2}-\frac{42}{144}\cdot25-5\[2pt] &=42\left(x-\frac{5}{12}\right)^{2}-\frac{7}{24}\cdot25-5\[2pt] &=42\left(x-\frac{5}{12}\right)^{2}-\frac{175}{24}-5. \end{aligned} ]

Since (-5 = -\frac{120}{24}),

[ 42\left(x-\frac{5}{12}\right)^{2}-\frac{175}{24}-\frac{120}{24} =42\left(x-\frac{5}{12}\right)^{2}-\frac{295}{24}. ]

Thus

Thus, the vertex form of the quadratic is

[ 42\left(x-\frac{5}{12}\right)^{2}-\frac{295}{24}, ]

which reveals the vertex of the parabola at (\left(\frac{5}{12},,-\frac{295}{24}\right)) and confirms that the graph opens upward.

Conclusion

We have thoroughly analyzed the quadratic expression (42x^{2}-35x-5). After verifying that no greatest common factor exists beyond 1 and that the trinomial is not factorable over the integers (since the discriminant (\Delta = 2065) is not a perfect square), we employed the quadratic formula to find the exact irrational roots (\frac{35\pm\sqrt{2065}}{84}). Finally, we rewrote the expression in vertex form by completing the square, identifying the vertex and the direction of the parabola. Whether the goal is solving the corresponding equation, graphing the function, or simply simplifying the expression, these tools—factoring checks, the quadratic formula, and completing the square—provide a complete algebraic toolkit for handling any quadratic that resists elementary factoring Most people skip this — try not to..