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Chapter 3: Problem 76

Find the zeros of the function algebraically. Give exact answers. $$f(x)=x^{2}-3 x-7$$

Short Answer

Expert verified
The zeros are \(x = \frac{3 + \sqrt{37}}{2}\) and \(x = \frac{3 - \sqrt{37}}{2}\).

Step by step solution

01

- Identify the components of the quadratic equation

Recognize that the function is a quadratic equation of the form \(ax^2 + bx + c=0\). Here, \(a=1\), \(b=-3\), and \(c=-7\).
02

- Use the quadratic formula

The quadratic formula is given by: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]. Plug in the values of \(a\), \(b\), and \(c\).
03

- Calculate the discriminant

First, calculate the discriminant \(b^2 - 4ac\): \((-3)^2 - 4(1)(-7) = 9 + 28 = 37\).
04

- Solve for the zeros using the quadratic formula

Substitute the values into the quadratic formula: \(x = \frac{3 \pm \sqrt{37}}{2}\).

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Key Concepts

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

quadratic equation
A quadratic equation is a polynomial equation of the form: \(ax^2 + bx + c = 0\). In this standard form:
  • \(a\) is the coefficient of \(x^2\) (the quadratic term),
  • \(b\) is the coefficient of \(x\) (the linear term), and
  • \(c\) is the constant term.

To find the zeros of a quadratic function means to determine the values of x for which the function equals zero, \(f(x) = 0\).
In the given example, the quadratic function provided is \(f(x) = x^2 - 3x - 7\).
In this case:
  • \(a = 1\)
  • \(b = -3\)
  • \(c = -7\)

Recognizing the values of \(a, b\), and \(c\) is the crucial first step to solving the quadratic equation.
quadratic formula
The quadratic formula is a powerful tool for finding the zeros of any quadratic function. The formula is given by:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
This formula will give you the exact roots of a quadratic equation, as long as you know the values of \(a, b\), and \(c\).
In the example function, we substitute the known values into the quadratic formula:
  • \(a = 1\)
  • \(b = -3\)
  • \(c = -7\)

Plugging these values in, the equation becomes:
\[x = \frac{3 \pm \sqrt{(-3)^2 - 4(1)(-7)}}{2(1)}\]
discriminant calculation
The discriminant is the part of the quadratic formula underneath the square root:
\(b^2 - 4ac\).
It is crucial because it tells us about the nature of the roots of the quadratic equation.
The discriminant can help us predict whether the roots are:
  • Real and distinct (if the discriminant is positive)
  • Real and equal (if the discriminant is zero)
  • Non-real complex numbers (if the discriminant is negative)

For our function, calculate the discriminant:
\[(-3)^2 - 4(1)(-7) = 9 + 28 = 37\]
Since the discriminant (37) is greater than zero, the quadratic function has two distinct real physical roots.
algebraic solving
After calculating the discriminant, we substitute back into the quadratic formula to solve for \(x\). Using the values from our example:
\[x = \frac{3 \pm \sqrt{37}}{2}\]
This results in two solutions because of the \( \pm \) symbol:
\[x_1 = \frac{3 + \sqrt{37}}{2}\]
\[x_2 = \frac{3 - \sqrt{37}}{2}\]
These two solutions are the zeros or roots of the quadratic function \(f(x) = x^2 - 3x - 7\).
By using the quadratic formula and calculating the discriminant, we have found the exact values where the function equals zero.

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