Can Plugging in (x/2) or (x - 1/2) Determine Real Zeros in a Quadratic Equation?

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In summary, the conversation discusses replacing f(x) with x - x^2 and plugging in different values, as well as the range and shifts of the graph of $f(x)$. It also mentions the concavity and discriminant of quadratic functions.
  • #1
mathland
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I can replace f(x) with x - x^2. Should I plug (x/2) into f(x)? How about (x - 1/2) into f(x)? I need the set up.

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  • #2
Note the range of $f(x)$. A and B are vertical shifts, C and D are horizontal shifts. Which one of the given shifts would result in the graph of $f(x)$ not crossing the $x-\text{axis}$?
 
  • #3
Greg said:
Note the range of $f(x)$. A and B are vertical shifts, C and D are horizontal shifts. Which one of the given shifts would result in the graph of $f(x)$ not crossing the $x-\text{axis}$?

Let y = the function.

y = x - x^2 - 1/2 does not cross the line y = 0.
 
  • #4
Note the concavity of $f(x)$. What does that tell you about lowering the graph of $f(x)$? (the line $y=0$ does not concern us presently).
 
  • #5
write each quadratic in standard form, $ax^2+bx+c$ ... check each discriminant, $D = b^2-4ac$

you know what the discriminant can tell you about the nature of zeros, right?
 

1. What is a quadratic with no real zeros?

A quadratic with no real zeros is a type of quadratic equation where the solutions or roots are imaginary numbers, meaning they involve the square root of a negative number. This type of quadratic cannot be solved using real numbers.

2. How do you know if a quadratic has no real zeros?

A quadratic has no real zeros if its discriminant, which is the expression under the square root in the quadratic formula, is negative. This means that the solutions will involve imaginary numbers.

3. Can a quadratic with no real zeros have any solutions?

Yes, a quadratic with no real zeros can still have solutions, but they will involve imaginary numbers. These solutions can be expressed in the form of a+bi, where a and b are real numbers and i is the imaginary unit.

4. How do you graph a quadratic with no real zeros?

To graph a quadratic with no real zeros, you can plot points using the x and y values from the solutions involving imaginary numbers. These points will be located on the imaginary axis of the coordinate plane.

5. What are some real-life applications of quadratics with no real zeros?

Quadratics with no real zeros are commonly used in engineering and physics, particularly in the study of oscillations and vibrations. They can also be used in financial modeling, such as in predicting stock prices or analyzing population growth.

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