Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 10: Problem 15
\(2 x^{2}+8 x-3=0\)
Short Answer
Expert verified
The equation has two solutions: \(x = \frac{-4 + \sqrt{22}}{2}\) and \(x = \frac{-4 - \sqrt{22}}{2}\).
Step by step solution
01
Identify the Quadratic Equation
The given equation is a quadratic equation in the standard form of \(ax^2 + bx + c = 0\) where \(a = 2\), \(b = 8\), and \(c = -3\). We will use the quadratic formula to solve for \(x\).
02
Apply the Quadratic Formula
The quadratic formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). In this equation, substitute \(a = 2\), \(b = 8\), and \(c = -3\) to find the solutions for \(x\).
03
Calculate the Discriminant
The discriminant is \(b^2 - 4ac\). For the equation, it is \(8^2 - 4 \times 2 \times (-3) = 64 + 24 = 88\). A positive discriminant indicates two real and distinct solutions.
04
Compute the Roots
Using the quadratic formula, substitute the values into \(x = \frac{-8 \pm \sqrt{88}}{4}\). This results in two solutions for \(x\): \(x = \frac{-8 + \sqrt{88}}{4}\) and \(x = \frac{-8 - \sqrt{88}}{4}\).
05
Simplify the Square Root
The square root of 88 can be further simplified to \(2\sqrt{22}\). The solutions then become \(x = \frac{-8 + 2\sqrt{22}}{4}\) and \(x = \frac{-8 - 2\sqrt{22}}{4}\).
06
Simplify the Solutions
Divide each part of the solution by 4: \(x = \frac{-4 + \sqrt{22}}{2}\) and \(x = \frac{-4 - \sqrt{22}}{2}\). These are the simplified forms of the roots.
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 for finding solutions to any quadratic equation of the form: \(ax^2 + bx + c = 0\). It helps us find the roots or solutions for \(x\) by using the coefficient values from the equation. The formula is:
- \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
- "\(a\)" stands for the coefficient of \(x^2\).
- "\(b\)" is the coefficient of \(x\).
- "\(c\)" is the constant term, the number without \(x\).
Discriminant
The discriminant is a crucial part of the quadratic formula and is represented by \(b^2 - 4ac\). It helps determine the nature and number of the roots of the quadratic equation. Here's what you can tell from the discriminant:
- If \(b^2 - 4ac > 0\), there are two distinct real roots. This means the equation has two different solutions.
- If \(b^2 - 4ac = 0\), there is exactly one real root, often called a repeated or double root. This happens when both solutions are essentially the same.
- If \(b^2 - 4ac < 0\), there are no real roots. Here, the roots are complex numbers and are not along the real number line.
Roots of Quadratic Equations
The roots of a quadratic equation are the values of \(x\) that satisfy the equation \(ax^2 + bx + c = 0\). Simply, these are the points where the graph of the equation crosses the x-axis. Using the quadratic formula, we find these roots by solving for \(x\). In our example with the equation \(2x^2 + 8x - 3 = 0\), after using the quadratic formula, the roots are calculated as:
- \(x = \frac{-4 + \sqrt{22}}{2}\)
- \(x = \frac{-4 - \sqrt{22}}{2}\)
- The equation has two real solutions because of the positive discriminant determined earlier.
- Each root represents a point on the x-axis where the parabola touches or crosses it.
