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Chapter 3: Problem 77

Find the zeros of the function algebraically. Give exact answers. $$f(x)=x^{2}+2 x-5$$

Short Answer

Expert verified
The zeros are \(x = -1 + \sqrt{6}\) and \(x = -1 - \sqrt{6}\).

Step by step solution

01

Set the function equal to zero

To find the zeros of the function, set the function equal to zero:\(f(x) = x^2 + 2x - 5 = 0\)
02

Apply the quadratic formula

The quadratic formula is used to find the zeros of a quadratic equation \(ax^2 + bx + c = 0\). The formula is:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]In this case, \(a = 1\), \(b = 2\), and \(c = -5\), so substitute these values into the formula.
03

Calculate the discriminant

First, calculate the discriminant \(b^2 - 4ac\):\[\text{Discriminant} = 2^2 - 4(1)(-5) = 4 + 20 = 24\]
04

Find the roots using the quadratic formula

Substitute the discriminant and the coefficients into the quadratic formula:\[x = \frac{-2 \pm \sqrt{24}}{2(1)}\]Simplify the square root of 24, which is \(2\sqrt{6}\):\[x = \frac{-2 \pm 2\sqrt{6}}{2}\]Simplify the expression further:\[x = -1 \pm \sqrt{6}\]
05

Write the exact zeros

The exact zeros of the function \(f(x) = x^2 + 2x - 5\) are:\(x = -1 + \sqrt{6}\)and\(x = -1 - \sqrt{6}\)

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Key Concepts

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

finding zeros
To find the zeros of a function means to determine the values of x where the function equals zero. These are also known as the roots or solutions of the equation. For the quadratic function given, we start by setting the equation equal to zero:
df(x) = x^2 + 2x - 5 = 0d
This step converts the problem into a standard quadratic equation form. Once in this form, we can use various methods to find the zeros, with the quadratic formula being one of the most reliable. The zeros of a function are the points where the graph of the function crosses the x-axis. In our example, the zeros are the exact values of x such as x = -1 + 鈭�6 and x = -1 - 鈭�6.
quadratic equations
A quadratic equation is an equation of the form ax^2 + bx + c = 0, where a, b, and c are constants, and a 鈮� 0. In our case, the equation is
x^2 + 2x - 5 = 0
The general form highlights three important coefficients:
  • a (the coefficient of x^2)
  • b (the coefficient of x)
  • c (the constant term)
These coefficients determine the shape and position of the parabola represented by the equation. For example, our quadratic equation has a=1, b=2, and c=-5. Quadratic equations can have two real roots, one real root, or no real roots, depending on the value of the discriminant.
discriminant
The discriminant of a quadratic equation provides important information about the nature of its roots. It is calculated using the formula:
b^2 - 4ac
In our quadratic equation where a=1, b=2, and c=-5, the discriminant is calculated as:
2^2 - 4(1)(-5) = 4 + 20 = 24
The discriminant can tell us the following:
  • If the discriminant is positive, there are two distinct real roots.
  • If the discriminant is zero, there is exactly one real root.
  • If the discriminant is negative, there are no real roots (the roots are complex).
In our example, the discriminant is 24 (positive), indicating that there are two distinct real roots. These roots are found using the quadratic formula, leading to the solutions x = -1 + 鈭�6 and x = -1 - 鈭�6.

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