/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 176 In the following exercises, find... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 176

In the following exercises, find the \(x\) - and \(y\) -intercepts. $$ y=x^{2}+6 x+13 $$

Short Answer

Expert verified
The y-intercept is \((0, 13)\) and the x-intercepts are complex: \(x = -3 \pm 2i\).

Step by step solution

01

Find the y-intercept

To find the y-intercept, set \(x = 0\) and solve for \(y\). Substitute \(x = 0\) into \(y = x^2 + 6x + 13\), resulting in: \(y = 0^2 + 6(0) + 13 = 13\). So, the y-intercept is \(13\), or the point \((0, 13)\).
02

Find the x-intercepts

To find the x-intercepts, set \(y = 0\) and solve for \(x\). So, \(0 = x^2 + 6x + 13\). This equation can be solved using the quadratic formula \(x = \frac {-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\), \(b = 6\), and \(c = 13\).
03

Apply the quadratic formula

Substitute the values of \(a\), \(b\), and \(c\) into the quadratic formula: \(x = \frac {-6 \pm \sqrt{6^2 - 4(1)(13)}}{2(1)} = \frac {-6 \pm \sqrt{36 - 52}}{2}\). This simplifies to: \(x = \frac {-6 \pm \sqrt{-16}}{2}\).
04

Simplify the solution

Since \(\sqrt{-16} = 4i\) (where \(i\) is the imaginary unit), the equation becomes: \(x = \frac {-6 \pm 4i}{2} = -3 \pm 2i\). Thus, the x-intercepts are complex numbers: \(x = -3 + 2i\) and \(x = -3 - 2i\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

y-intercept
The y-intercept of a function is where the graph crosses the y-axis. To find it, set the value of x to zero and solve for y. For example, with the function \( y = x^2 + 6x + 13 \), we substitute x with 0. This gives us:
  • \( y = 0^2 + 6(0) + 13 \)
  • \( y = 13 \)
So, the y-intercept is at the point (0, 13). This tells us that when x is zero, y equals 13. This point is important because it helps us understand where the graph meets the y-axis.
x-intercepts
The x-intercepts are the points where the graph crosses the x-axis. To find them, set y to zero and solve for x. For our example, we start with:
  • \( 0 = x^2 + 6x + 13 \)
Now, we need to solve this quadratic equation. The solution might involve real numbers or complex numbers, depending on the discriminant \( b^2 - 4ac \). Let's use the quadratic formula to find the x-intercepts in more detail.
quadratic formula
The quadratic formula is used to solve quadratic equations of the form \( ax^2 + bx + c = 0 \). The formula is given by: \[ x = \frac {-b \pm \sqrt{b^2 - 4ac}}{2a} \] In our case, \( a = 1 \), \( b = 6 \), and \( c = 13 \). Substituting these into the formula, we get: \[ x = \frac {-6 \pm \sqrt{6^2 - 4(1)(13)}}{2(1)} = \frac {-6 \pm \sqrt{36 - 52}}{2} \] This simplifies to: \[ x = \frac {-6 \pm \sqrt{-16}}{2} \] Because the expression under the square root is negative, the solutions involve complex numbers.
complex numbers
Complex numbers arise when we take the square root of a negative number. The imaginary unit \( i \) is defined as \( \sqrt{-1} \). Thus, \( \sqrt{-16} = 4i \). Using this in our quadratic formula, we have:
  • \( x = \frac {-6 \pm 4i}{2} = -3 \pm 2i \)
So, the x-intercepts are actually complex numbers: \( -3 + 2i \) and \( -3 - 2i \). These points do not cross the x-axis on a real number plane, but they are crucial in understanding the complete behavior of quadratic functions, including those that do not have real-number solutions. Complex numbers are essential for solving equations where real numbers are not sufficient.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Determine the number of solutions to each quadratic equation. (a) \(9 v^{2}-15 v+25=0\) \(100 w^{2}+60 w+9=0\) \(5 c^{2}+7 c-10=0\) \(15 d^{2}-4 d+8=0\)

In the following exercises, solve. Round answers to the nearest tenth. A stone is thrown vertically upward from a platform that is 20 feet high at a rate of \(160 \mathrm{ft} / \mathrm{sec}\). Use the quadratic equation \(h=-16 t^{2}+160 t+20\) to find how long it will take the stone to reach its maximum height, and then find the maximum height.

In the following exercises, graph by using intercepts, the vertex, and the axis of symmetry. $$ y=x^{2}+6 x+5 $$

In the following exercises, find (a) the axis of symmetry and (b) the vertex. $$ y=-x^{2}+2 x+5 $$

Solve by using the Quadratic Formula. \(p^{2}-6 p-27=0\)
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.