/*! 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 33 \(x^{2}+4 x-9=0\)... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 33

\(x^{2}+4 x-9=0\)

Short Answer

Expert verified
The solutions are \(-2 + \sqrt{13}\) and \(-2 - \sqrt{13}\).

Step by step solution

01

Identify the coefficients

In the quadratic equation \(ax^2 + bx + c = 0\), identify the coefficients: \(a = 1, b = 4, c = -9\).
02

Calculate the Discriminant

Use the discriminant formula \(D = b^2 - 4ac\) to find the discriminant. \(D = 4^2 - 4 \cdot 1 \cdot (-9) = 16 + 36 = 52\).
03

Apply the Quadratic Formula

The quadratic formula is \(x = \frac{-b \pm \sqrt{D}}{2a}\). Substitute the values: \(a = 1, b = 4, D = 52\).
04

Solve for the roots

Using the quadratic formula, \(x = \frac{-4 \pm \sqrt{52}}{2 \cdot 1} = \frac{-4 \pm 2 \sqrt{13}}{2} = -2 \pm \sqrt{13}\).

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.

quadratic formula
The **quadratic formula** is a powerful tool used to find the roots of any quadratic equation. Quadratic equations are in the form \(ax^2 + bx + c = 0\). To solve this, the quadratic formula is used: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]. This formula works for any values of \(a\), \(b\), and \(c\) as long as \(a \eq 0\).
In our exercise, we applied the quadratic formula to the equation \(x^2 + 4x - 9 = 0\). By substituting the identified coefficients \(a = 1\), \(b = 4\), and \(c = -9\), we were able to determine the roots efficiently.
discriminant
The **discriminant** is a component of the quadratic formula that determines the nature of the roots. The discriminant is given by the expression \(D = b^2 - 4ac\).
The value of the discriminant reveals important information:
  • **If D > 0**: There are two distinct real roots.
  • **If D = 0**: There is exactly one real root.
  • **If D < 0**: There are two complex roots.

In our specific case, the discriminant calculation was \(D = 4^2 - 4 \cdot 1 \cdot (-9) = 52\). Since \(52 > 0\), we confirmed that there are two distinct real roots for the equation \(x^2 + 4x - 9 = 0\).
roots of quadratic equations
Finding the **roots of quadratic equations** is essential to solving these types of problems. The roots are the values of \(x\) that satisfy the equation \(ax^2 + bx + c = 0\). Using the quadratic formula, we can solve for these roots.
For the quadratic equation \(x^2 + 4x - 9 = 0\), we substituted the values into the formula: \[ x = \frac{-4 \pm \sqrt{52}}{2 \cdot 1} = \frac{-4 \pm 2 \sqrt{13}}{2} \].
Simplifying, we found the roots to be \(-2 \pm \sqrt{13}\). These are the values of \(x\) that solve the original quadratic equation.
coefficients in quadratic equations
The **coefficients in quadratic equations** play a crucial role in determining the behavior and solutions of the equation. In the equation \(ax^2 + bx + c = 0\),
  • \

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

\(k^{2}=-72\)

\(2 n^{2}+6=-20\)

Scale is removed from steel before galvanizing with an \(18 \%\) hydrochloric acid solution. Hydrochloric acid solution is delivered by rail in a tanker. This solution is \(31.45 \%\) hydrochloric acid. Find the amount of water and the amount of acid solution needed to make 30,000 gallons of a new solution that is \(18 \%\) hydrochloric acid. Round to the nearest whole number.

\(z^{2}+3 z+14=0\)

Is the graph of \(y=5 x^{2}\) shorter or taller than the graph of \(y=3 x^{2}\) ? Explain.
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Decision 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.