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Magazine Chapter 10: Problem 18
\(3 x^{2}-6 x+2=0\)
Short Answer
Expert verified
The solutions are \(x = 1 \pm \frac{\sqrt{3}}{3}\).
Step by step solution
01
Identify the Quadratic Equation
We start with the quadratic equation given by \(3x^2 - 6x + 2 = 0\). This is in the standard form \(ax^2 + bx + c = 0\) where \(a = 3\), \(b = -6\), and \(c = 2\).
02
Calculate the Discriminant
The discriminant \(D\) of a quadratic equation \(ax^2 + bx + c = 0\) is calculated using the formula \(D = b^2 - 4ac\). Substituting the values of \(a\), \(b\), and \(c\):\[D = (-6)^2 - 4(3)(2) = 36 - 24 = 12\]The discriminant is 12.
03
Determine the Nature of the Roots
Since the discriminant \(D = 12\) is greater than zero, the quadratic equation has two distinct real roots.
04
Use the Quadratic Formula to Find the Roots
The solutions to the quadratic equation \(ax^2 + bx + c = 0\) can be found using the quadratic formula:\[x = \frac{{-b \pm \sqrt{D}}}{2a}\]Substitute \(a = 3\), \(b = -6\), and \(D=12\):\[x = \frac{{6 \pm \sqrt{12}}}{6}\]Simplify \(\sqrt{12} = 2\sqrt{3}\):\[x = \frac{{6 \pm 2\sqrt{3}}}{6}\]Simplify further by dividing all terms by 2:\[x = \frac{{3 \pm \sqrt{3}}}{3}\]Hence, the roots are \(x = 1 + \frac{\sqrt{3}}{3}\) and \(x = 1 - \frac{\sqrt{3}}{3}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding the Discriminant
In quadratic equations, the discriminant plays a critical role in determining the nature and number of roots. The discriminant, often represented as \(D\), is the part of the quadratic formula under the square root: \(D = b^2 - 4ac\). It's derived from the coefficients \(a\), \(b\), and \(c\) in the standard quadratic equation \(ax^2 + bx + c = 0\).
The value of the discriminant tells us about the roots of the quadratic equation. Here's what it can reveal:
The value of the discriminant tells us about the roots of the quadratic equation. Here's what it can reveal:
- If \(D > 0\), the equation has two distinct real roots.
- If \(D = 0\), the equation has exactly one real root, which is sometimes called a double root.
- If \(D < 0\), the equation has no real roots but two complex roots.
Using the Quadratic Formula
The quadratic formula is a powerful tool used to find the roots of any quadratic equation of the form \(ax^2 + bx + c = 0\). The formula is expressed as:\[x = \frac{{-b \pm \sqrt{D}}}{2a}\]where \(D\) is the discriminant \(b^2 - 4ac\).
This formula allows you to solve any quadratic equation by simply substituting the values of \(a\), \(b\), and \(c\). Here's what you do:
This formula allows you to solve any quadratic equation by simply substituting the values of \(a\), \(b\), and \(c\). Here's what you do:
- Calculate the discriminant \(D\).
- Substitute \(a\), \(b\), and \(D\) into the formula.
- Simplify the expression under the square root.
- Perform the operations to find the values of \(x\).
Identifying Real Roots
Real roots are solutions to a quadratic equation that are real numbers. Whether or not a quadratic equation has real roots depends on the value of the discriminant \(D\). If \(D > 0\) or \(D = 0\), the quadratic equation will have real roots.
- Two distinct real roots occur when \(D > 0\).
- A single real root occurs when \(D = 0\), which means the vertex of the parabola touches the x-axis at a single point.
