Calcul 5x-5-x²
Use this premium interactive calculator to evaluate the quadratic expression 5x – 5 – x², inspect its vertex, understand whether a chosen value of x gives a positive or negative output, and visualize the curve instantly on a responsive chart.
Calculation Results
Enter a value and click Calculate to evaluate the expression.
Expert Guide to Calculating 5x – 5 – x²
The expression 5x – 5 – x² is a classic quadratic expression. If you are searching for “calcul 5x-5-x 2,” the intended task is typically to compute the value of the algebraic expression for a chosen number x, simplify it, and understand how it behaves as a graph. Written in standard mathematical notation, the expression is usually presented as 5x – 5 – x², which can also be rearranged as -x² + 5x – 5.
This is important because the standard form ax² + bx + c helps you identify the type of function immediately. In this case, a = -1, b = 5, and c = -5. Because the coefficient of x² is negative, the parabola opens downward. That means the function has a maximum point rather than a minimum point. Understanding this one feature helps you solve many practical school, exam, and graphing questions faster.
What does 5x – 5 – x² mean?
At a basic level, this expression combines three parts:
- 5x: a linear term that grows in direct proportion to x.
- -5: a constant term that shifts the whole graph downward by 5 units.
- -x²: a quadratic term that eventually dominates as x becomes large in magnitude and causes the graph to curve downward.
Because the x² term dominates for large positive and negative x, the overall value of the function decreases toward negative values at both ends. This is why the graph is a downward-opening parabola. If you evaluate the expression for several x values, you will see a rise, a peak, and then a fall.
How to calculate it step by step
To evaluate 5x – 5 – x², follow this process:
- Choose a value for x.
- Compute 5x.
- Compute x².
- Subtract 5.
- Subtract x² from the previous result.
For example, if x = 2:
- 5x = 5 × 2 = 10
- x² = 2² = 4
- 5x – 5 – x² = 10 – 5 – 4 = 1
If x = 3:
- 5x = 15
- x² = 9
- 15 – 5 – 9 = 1
If x = 1:
- 5x = 5
- x² = 1
- 5 – 5 – 1 = -1
This simple set of examples already hints at the shape of the parabola. The values rise from negative to positive, stay relatively high near the center, and then drop again.
Simplifying into standard form
Many teachers prefer to rewrite the expression as -x² + 5x – 5. This does not change the value at all. It only makes the function easier to analyze because the terms are in descending powers of x. Once in standard form, you can quickly identify:
- a = -1 so the graph opens downward
- b = 5 so the axis of symmetry is x = -b / 2a = 2.5
- c = -5 so the y-intercept is -5
Finding the vertex
The vertex of a quadratic function in standard form is found using the formula:
x = -b / (2a)
For -x² + 5x – 5:
- a = -1
- b = 5
- x = -5 / (2 × -1) = 2.5
Now substitute x = 2.5 into the expression:
- 5(2.5) – 5 – (2.5)²
- 12.5 – 5 – 6.25
- = 1.25
So the vertex is (2.5, 1.25). Since the parabola opens downward, this is the maximum point of the graph.
Roots and where the expression equals zero
Another useful question is when 5x – 5 – x² = 0. Rearranging gives:
x² – 5x + 5 = 0
Apply the quadratic formula:
x = (5 ± √(25 – 20)) / 2 = (5 ± √5) / 2
This produces approximate roots:
- x ≈ 1.382
- x ≈ 3.618
That tells you the expression is positive only between those two x-values, and negative outside them. This is an excellent shortcut for sign analysis.
Sample value table for 5x – 5 – x²
| x | 5x | x² | 5x – 5 – x² |
|---|---|---|---|
| -1 | -5 | 1 | -11 |
| 0 | 0 | 0 | -5 |
| 1 | 5 | 1 | -1 |
| 2 | 10 | 4 | 1 |
| 2.5 | 12.5 | 6.25 | 1.25 |
| 3 | 15 | 9 | 1 |
| 4 | 20 | 16 | -1 |
| 5 | 25 | 25 | -5 |
This table shows a very clear pattern. The values increase up to x = 2.5 and then decrease. It also reveals a symmetry: the outputs at x = 2 and x = 3 are equal, and the outputs at x = 1 and x = 4 are equal. That symmetry always occurs around the axis x = 2.5.
Why quadratics matter in real education and careers
Understanding expressions like 5x – 5 – x² is not just an academic exercise. Quadratic functions appear in physics, finance, engineering design, optimization, statistics, and computer graphics. In education, algebra proficiency is a major predictor of readiness for more advanced mathematics and technical study. Government and university data consistently show that stronger mathematics skills are linked with broader opportunity.
| Education or labor metric | Statistic | Source |
|---|---|---|
| U.S. grade 8 students at or above NAEP Proficient in mathematics | Approximately 26% | National Center for Education Statistics, NAEP |
| Median weekly earnings for workers with a bachelor’s degree | Higher than the all-worker median | U.S. Bureau of Labor Statistics |
| STEM occupations typical wage premium | Often above non-STEM median levels | U.S. Bureau of Labor Statistics |
The exact values vary by year, but the trend is stable: quantitative skills support academic and economic opportunity. If you are practicing with quadratic expressions now, you are building a foundation for future coursework in calculus, data science, economics, and engineering.
Comparison: linear thinking versus quadratic thinking
One of the biggest challenges for students is moving from linear to quadratic reasoning. A linear expression changes by a constant amount. A quadratic expression changes by a varying amount because the squared term grows faster and faster. The table below highlights the conceptual difference.
| Feature | Linear expression, example 5x – 5 | Quadratic expression, example 5x – 5 – x² |
|---|---|---|
| Highest power of x | 1 | 2 |
| Graph shape | Straight line | Parabola |
| Rate of change | Constant | Not constant |
| Extremum | Usually none | Has a maximum here because a is negative |
| Symmetry | No axis symmetry in general | Symmetric about x = 2.5 |
Common mistakes when evaluating 5x – 5 – x²
- Forgetting order of operations. You should square x before subtracting it.
- Confusing -x² with (-x)². These are not the same in every context. The expression means the negative of x squared.
- Dropping the negative sign. Writing x² instead of -x² changes the entire graph.
- Mixing up standard form and factor form. Always verify the coefficients before using formulas.
- Assuming the graph opens upward. Here, a = -1, so the graph opens downward.
How to check your answer quickly
There are several fast ways to validate a result:
- Substitute the x value twice, once directly and once after rewriting as -x² + 5x – 5.
- Compare your result to nearby values. If x is near 2.5, your answer should be near the maximum of 1.25.
- Use the graph. If your point seems far away from the plotted curve, reevaluate your arithmetic.
- Check signs carefully. The most common error is turning -x² into +x².
How this calculator helps
The calculator above does more than return a single number. It also estimates the vertex, tells you whether the selected x-value produces a positive or negative result, and plots the entire quadratic curve around your chosen point. This visual feedback is valuable because many learners understand quadratics faster when they can connect formulas, numerical substitution, and graph behavior in one place.
By adjusting the graph range and step size, you can inspect both the broad shape of the parabola and the local behavior near roots or near the vertex. For example, selecting a smaller step like 0.25 makes the curve denser and can help you spot the peak more clearly around x = 2.5.
Trusted educational references
If you want to explore algebra standards, math achievement data, and STEM outcomes further, these authoritative sources are useful:
- National Center for Education Statistics (NCES) mathematics assessment data
- U.S. Bureau of Labor Statistics occupational outlook for math-related fields
- OpenStax College Algebra from Rice University
Final takeaway
To solve “calcul 5x-5-x 2,” think of the expression as y = 5x – 5 – x² or equivalently y = -x² + 5x – 5. Evaluate it by substitution, analyze it as a quadratic, and remember the most important facts: it opens downward, its vertex is at (2.5, 1.25), its y-intercept is -5, and it crosses the x-axis near 1.382 and 3.618. Once you understand those features, you are not just calculating a value. You are reading the entire behavior of the function.