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Chapter 0: Problem 4

Use the quadratic formula to find the zeros of the functions. $$f(x)=\frac{1}{4} x^{2}+x+1$$

Short Answer

Expert verified
The zero of the function is \( x = -2 \).

Step by step solution

01

Identify Coefficients

For the quadratic equation in the standard form, identify the coefficients: \ The given function is \[ f(x) = \frac{1}{4}x^2 + x + 1 \] \( a = \frac{1}{4} \) \( b = 1 \) \( c = 1 \)
02

Write Down the Quadratic Formula

The quadratic formula to find the zeros of the function is: \[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} \]
03

Substitute into the Quadratic Formula

Substitute the identified coefficients into the quadratic formula: \[ x = \frac{{-1 \pm \sqrt{{1^2 - 4 \cdot \frac{1}{4} \cdot 1}}}}{{2 \cdot \frac{1}{4}}} \] This simplifies to: \[ x = \frac{{-1 \pm \sqrt{{1 - 1}}}}{{0.5}} \] \[ x = \frac{{-1 \pm \sqrt{0}}}{{0.5}} \]
04

Solve the Equation

Since the discriminant is 0, the equation simplifies to: \[ x = \frac{{-1}}{0.5} \] \[ x = -2 \]
05

Conclusion

Thus, the quadratic function \[ f(x)=\frac{1}{4} x^{2}+x+1 \] has one real root: \[ x = -2 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

zeros of a function
The zeros of a function are the values of x for which the function equals zero. When dealing with quadratic functions, these zeros are also referred to as roots or solutions of the equation. For a function like \[ f(x) = \frac{1}{4} x^2 + x + 1 \], finding the zeros involves solving the equation \[ \frac{1}{4} x^2 + x + 1 = 0 \]. This process helps us understand where the graph of the function intersects the x-axis. Solving for zeros is crucial because it provides insights into the behavior of the function, such as where it changes signs.
discriminant
The discriminant is a key part of the quadratic formula, recognizable in the expression under the square root: \[ b^2 - 4ac \]. It helps determine the nature of the roots of a quadratic equation. Depending on its value, we can identify:

  • If the discriminant is positive, the equation has two distinct real roots.
  • If it is zero, there is exactly one real root, as seen in our example with the function \[ f(x) = \frac{1}{4} x^2 + x + 1 \], which simplified to give \[ x = -2 \].
  • If the discriminant is negative, the equation has no real roots, only complex ones.
This makes the discriminant a powerful tool for predicting the nature of the solutions without actually solving the equation.
solving quadratic equations
To solve quadratic equations, one common method is using the quadratic formula: \[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} \]. This formula solutions are derived from the coefficients of the given quadratic function in the form \[ ax^2 + bx + c = 0 \]. Steps include:

  • Identify the coefficients a, b, and c from the equation.
  • Substitute these values into the quadratic formula.
  • Calculate the discriminant \[ b^2 - 4ac \] and determine the nature of the roots.
  • Solve for x by simplifying the expression.
In our example, substituting \[ a = \frac{1}{4}, b = 1, c = 1 \] into the quadratic formula, we found the root to be \[ x = -2 \], confirming the equation has a single real solution.
quadratic function
A quadratic function is a second-degree polynomial function of the form \[ f(x) = ax^2 + bx + c \], where a, b, and c are constants and a 鈮� 0. These functions have distinctive U-shaped graphs called parabolas, which can open upward or downward based on the sign of the coefficient a. Key features of quadratic functions include:

  • The vertex, representing the highest or lowest point on the graph.
  • The axis of symmetry, a vertical line through the vertex.
  • The direction of the parabola's opening determined by the sign of a.
Understanding quadratic functions is important in various fields, including physics, engineering, and economics, as they model many natural phenomena and technical processes.

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Most popular questions from this chapter

When a car's brakes are slammed on at a speed of \(x\) miles per hour, the stopping distance is \(\frac{1}{20} x^{2}\) feet. Show that when the speed is doubled the stopping distance increases fourfold.

Assume that a \(\$ 1000\) investment earns interest compounded semiannually. Express the value of the investment after 2 years as a polynomial in the annual rate of interest \(r.\)

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