To find an expression equivalent to 2x² + 2x + 7, we can analyze the polynomial and look for ways to factor or rewrite it.
Firstly, let’s rewrite the given expression clearly:
2x² + 2x + 7
We can see that this is a quadratic polynomial in the format of ax² + bx + c, where:
- a = 2
- b = 2
- c = 7
This polynomial does not factor neatly into simpler binomials due to the presence of the constant term 7, which is not easily divisible by the coefficients of x.
However, we can also express it as:
2(x² + x) + 7
This representation highlights the quadratic and linear components while still maintaining the integrity of the original expression.
To further explore this expression, we can complete the square:
2(x² + x + (1/4) - (1/4)) + 7
Since we add and subtract (1/4) inside the parentheses:
2((x + 1/2)² - 1/4) + 7
Now simplifying gives us:
2(x + 1/2)² - 1/2 + 7
This simplifies to:
2(x + 1/2)² + 13/2
So, an equivalent expression to 2x² + 2x + 7 can be expressed as:
2(x + 1/2)² + 13/2
This form emphasizes the vertex of the parabola represented by this quadratic, which can be particularly useful for graphing or further analysis.
In conclusion, the expression 2x² + 2x + 7 can be rewritten in multiple equivalent forms. One such form is 2(x + 1/2)² + 13/2.