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Chapter 11: Problem 50

Find any \(x\) -intercepts and the \(y\) -intercept. If no \(x\) -intercepts exist, state this. $$ f(x)=x^{2}+5 x+2 $$

Short Answer

Expert verified
The y-intercept is (0, 2). The x-intercepts are \( \left( \frac{-5 + \sqrt{17}}{2}, 0 \right) \) and \( \left( \frac{-5 - \sqrt{17}}{2}, 0 \right) \).

Step by step solution

01

Identify the Equation

The given function is \(f(x) = x^2 + 5x + 2 \).
02

Find the y-intercept

To find the y-intercept, set \(x = 0\) and solve for \(f(0)\). \[ f(0) = 0^2 + 5(0) + 2 = 2 \] So the y-intercept is \((0, 2)\).
03

Find the x-intercepts

To find the x-intercepts, set \(f(x) = 0\) and solve for \(x\). This means solving the equation: \[ x^2 + 5x + 2 = 0 \].
04

Apply the Quadratic Formula

Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \(a = 1\), \(b = 5\), and \(c = 2\).
05

Calculate the Discriminant

Calculate the discriminant \(\Delta = b^2 - 4ac\): \[ \Delta = 5^2 - 4(1)(2) = 25 - 8 = 17 \]
06

Solve for x-intercepts

Since the discriminant is positive, there are two real solutions. Substitute the values into the quadratic formula: \[ x = \frac{-5 \pm \sqrt{17}}{2} \]
07

Simplify the Solutions

Simplify to find the solutions: \[ x = \frac{-5 + \sqrt{17}}{2} \] and \[ x = \frac{-5 - \sqrt{17}}{2} \]. So the x-intercepts are \( \left( \frac{-5 + \sqrt{17}}{2}, 0 \right) \) and \( \left( \frac{-5 - \sqrt{17}}{2}, 0 \right) \).

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Key Concepts

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

Exploring x-intercepts
Finding the x-intercepts of a quadratic function involves solving the equation for points where the graph crosses the x-axis. These points are essential as they help to understand the roots or solutions of the quadratic equation. To find the x-intercepts for the equation given, set the function equal to zero and solve for x. In our problem, the quadratic equation is: \[ x^2 + 5x + 2 = 0 \]. We use the quadratic formula, which we will discuss in detail later. By solving this equation, we find our x-intercepts at: \[ x = \frac{-5 + \sqrt{17}}{2} \] and \[ x = \frac{-5 - \sqrt{17}}{2} \]. These x-intercepts tell us where the parabola crosses the x-axis.
Understanding the y-intercept
To find the y-intercept, we look for the point where the graph of the function crosses the y-axis. This is achieved by setting x=0 in your quadratic equation. For our function, \[ f(x) = x^2 + 5x + 2 \], we calculate: \[ f(0) = 0^2 + 5(0) + 2 = 2 \]. This results in a y-intercept at the point (0, 2). The y-intercept is crucial for graphing because it tells us the initial value of the function when x equals zero.
Quadratic Formula Demystified
The quadratic formula is a powerful tool for solving quadratic equations of the form \( ax^2 + bx + c = 0 \). It provides the solutions for x by plugging in the coefficients a, b, and c. The formula is: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]. For our specific problem, a=1, b=5, and c=2. By substituting these values, the formula becomes: \[ x = \frac{-5 \pm \sqrt{25 - 8}}{2} \]. Simplify the discriminant to solve for x. The quadratic formula is essential because it guarantees finding the roots if they exist.
The Role of the Discriminant
The discriminant is a key part of the quadratic formula. It is represented by \( \Delta = b^2 - 4ac \). The discriminant helps in determining the nature of the roots of the quadratic equation:
  • If \( \Delta > 0 \), there are two distinct real roots.
  • If \( \Delta = 0 \), there is exactly one real root.
  • If \( \Delta < 0 \), there are no real roots, but two complex roots.
For our equation, \( \Delta = 25 - 8 = 17 \), which is greater than zero. Thus, we have two distinct real roots. Understanding the discriminant can quickly tell us how many real solutions our quadratic equation has.

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Most popular questions from this chapter

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