Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 10: Problem 109
Solve by using the Quadratic Formula. \(2 a^{2}-6 a+3=0\)
Short Answer
Expert verified
The solutions are \(a = \frac{3}{2} + \frac{\sqrt{3}}{2}\) and \(a = \frac{3}{2} - \frac{\sqrt{3}}{2}\).
Step by step solution
01
- Identify the coefficients
Identify the coefficients in the quadratic equation \(2a^2 - 6a + 3 = 0\). Here, the coefficient of \(a^2\) is \(a = 2\), the coefficient of \(a\) is \(b = -6\), and the constant term is \(c = 3\).
02
- Write down the Quadratic Formula
The Quadratic Formula is given by: \[ a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Plug in the values of \(a\), \(b\), and \(c\).
03
- Calculate the discriminant
Calculate the discriminant using the formula: \(b^2 - 4ac\). Substituting the given values: \((-6)^2 - 4(2)(3) = 36 - 24 = 12\).
04
- Find the square root of the discriminant
Calculate the square root of the discriminant: \(\sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3}\).
05
- Substitute and simplify
Substitute \(\sqrt{12}\) back into the Quadratic Formula: \[ a = \frac{-(-6) \pm 2\sqrt{3}}{2(2)} \] Simplify: \[ a = \frac{6 \pm 2\sqrt{3}}{4} = \frac{6}{4} \pm \frac{2\sqrt{3}}{4} = \frac{3}{2} \pm \frac{\sqrt{3}}{2} \].
06
- State the solutions
The solutions to the quadratic equation are: \[ a = \frac{3}{2} + \frac{\sqrt{3}}{2} \] and \[ a = \frac{3}{2} - \frac{\sqrt{3}}{2} \].
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 Equation
A quadratic equation is a type of polynomial equation involving a variable raised to the second power. It generally has the form: ax^2 + bx + c = 0, where a, b, and c are coefficients, and a 鈮� 0. Quadratic equations are essential because they appear in various mathematical contexts, and understanding how to solve them is a key algebraic skill. The quadratic term (ax^2), the linear term (bx), and the constant term (c) together shape the parabola described by the equation when graphed.
Let's solve the example given: 2a^2 - 6a + 3 = 0to see how these principles come together.
Let's solve the example given: 2a^2 - 6a + 3 = 0to see how these principles come together.
Discriminant
The discriminant is a crucial component of the quadratic formula, acting as a determinant of the nature and number of the roots for the equation. It is given by the expression [...theres nothing think better formula here than d = b^2 - 4ac,]
The value of the discriminant can reveal different properties of the roots:
The value of the discriminant can reveal different properties of the roots:
- If the discriminant d > 0, there are two distinct real roots.
- If d = 0, there is one real root (or a repeated root).
- If d < 0, the roots are complex or imaginary.
Roots of Quadratic Equations
The roots (or solutions) of a quadratic equation are the values of the variable that satisfy the equation. These can be found by solving the quadratic formula:
\[a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
In our example, the quadratic formula application yields:
\[a = \frac{-(-6) \pm 2\sqrt{3}}{2(2)}\]
Upon simplification:
\[a = \frac{6 \pm 2\sqrt{3}}{4} = \frac{3}{2} \pm \frac{\sqrt{3}}{2}\]
This unravels two specific roots: .
\[a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
In our example, the quadratic formula application yields:
\[a = \frac{-(-6) \pm 2\sqrt{3}}{2(2)}\]
Upon simplification:
\[a = \frac{6 \pm 2\sqrt{3}}{4} = \frac{3}{2} \pm \frac{\sqrt{3}}{2}\]
This unravels two specific roots: .
Algebraic Solutions
Algebraic solutions refer to solving equations using algebraic manipulations. For quadratic equations, the algebraic solution typically involves using the quadratic formula. Here is the step-by-step algebraic solution for our quadratic equation, 2a^2 - 6a + 3 = 0:
- Identify the coefficients: a = 2, b = -6, c = 3.
- Calculate the discriminant: (b^2 - 4ac) = 12.
- Compute the square root of the discriminant: \[\sqrt{12} = 2\sqrt{3}\]
- Substitute into the quadratic formula: \[a = \frac{6 \pm 2\sqrt{3}}{4}\]
- Simplify to find the roots: \[a = \frac{3}{2} \pm \frac{\sqrt{3}}{2}\] These algebraic steps systematically break down the problem to reach the solution, showing that anyone can master quadratic equations with practice and patience.
