Find The Discriminant Of 2x Square + X - 6 Is Equal To Zero?
Understanding how to find the discriminant of a quadratic equation is fundamental in algebra. It allows students and mathematicians to determine the nature of the roots without solving the equation explicitly. In this article, we will explore the process of calculating the discriminant for the quadratic equation 2x² + x - 6 = 0, explaining each step thoroughly. Whether you're preparing for exams, solving real-world problems, or just enhancing your mathematical skills, this guide will provide clarity and confidence in handling quadratic equations.
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What Is a Quadratic Equation?
Before diving into the discriminant, it's essential to understand the structure of quadratic equations.
Definition of a Quadratic Equation
A quadratic equation is a second-degree polynomial equation in a single variable, generally written in the form:\[ ax^2 + bx + c = 0 \]
where:
- \( a \), \( b \), and \( c \) are coefficients with \( a \neq 0 \),
- \( x \) is the variable.
In our specific example, the quadratic equation is:
\[ 2x^2 + x - 6 = 0 \]
Here:
- \( a = 2 \),
- \( b = 1 \),
- \( c = -6 \).
Why Is the Discriminant Important?
The discriminant helps us determine:
- Whether the quadratic equation has real roots,
- The nature of those roots (distinct or repeated),
- The number of solutions without solving the equation.
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Understanding the Discriminant
Definition of the Discriminant
The discriminant (\( D \)) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by the formula:\[ D = b^2 - 4ac \]
The value of \( D \) reveals the nature of the roots:
- If \( D > 0 \), the quadratic has two distinct real roots.
- If \( D = 0 \), the quadratic has exactly one real root (a repeated root).
- If \( D < 0 \), the quadratic has two complex conjugate roots.
Significance of the Discriminant's Value
Knowing the discriminant helps in:
- Quickly assessing the number of solutions,
- Determining whether solutions are real or complex,
- Planning for solving the quadratic more efficiently.
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Calculating the Discriminant of 2x² + x - 6 = 0
Let's proceed step-by-step to compute the discriminant for our specific quadratic equation.
Step 1: Identify the coefficients \( a \), \( b \), and \( c \)
From the equation:\[ 2x^2 + x - 6 = 0 \]
we have:
- \( a = 2 \),
- \( b = 1 \),
- \( c = -6 \).
Step 2: Apply the discriminant formula
Recall:
\[ D = b^2 - 4ac \]
Plugging in the values:
\[ D = (1)^2 - 4 \times 2 \times (-6) \]
Step 3: Simplify the expression
Calculate each component:- \( (1)^2 = 1 \),
- \( 4 \times 2 \times (-6) = 8 \times (-6) = -48 \).
\[ D = 1 - (-48) \]
which simplifies to:
\[ D = 1 + 48 = 49 \]
Step 4: Interpret the result
Since \( D = 49 \), which is greater than zero, we conclude:- The quadratic equation has two distinct real roots.
- The roots are real and different.
Implications of the Discriminant Result
Having found that \( D = 49 \), let's explore what this means practically.
Number of Roots
- The quadratic equation has two real roots because \( D > 0 \).
Nature of Roots
- The roots are distinct and real.
Calculating the Roots
Although the main focus is on the discriminant, knowing how to find the roots can be useful.The roots are given by the quadratic formula:
\[ x = \frac{-b \pm \sqrt{D}}{2a} \]
Applying this to our example:
- \( b = 1 \),
- \( D = 49 \),
- \( a = 2 \).
Calculate:
\[ x = \frac{-1 \pm \sqrt{49}}{2 \times 2} = \frac{-1 \pm 7}{4} \]
Thus, the two solutions are:
- \( x = \frac{-1 + 7}{4} = \frac{6}{4} = \frac{3}{2} \)
- \( x = \frac{-1 - 7}{4} = \frac{-8}{4} = -2 \)
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Additional Tips for Calculating Discriminants
Handling Different Coefficients
- Always identify the coefficients \( a \), \( b \), and \( c \) carefully.
- Be cautious with signs, especially when coefficients are negative.
Using Discriminant to Decide Next Steps
- If \( D > 0 \), proceed to find roots using the quadratic formula.
- If \( D = 0 \), there is one repeated root, which simplifies to \( x = -b/(2a) \).
- If \( D < 0 \), roots are complex and involve imaginary numbers.
Practice Examples
To master the concept, practice calculating the discriminant for various quadratic equations, such as:- \( 3x^2 - 4x + 1 = 0 \),
- \( -x^2 + 5x - 6 = 0 \),
- \( x^2 + 2x + 5 = 0 \).
Applications of the Discriminant in Real-World Problems
The discriminant isn't just an academic concept; it has practical applications across numerous fields.
Engineering and Physics
- Analyzing projectile motion where quadratic equations model trajectories.
- Determining stability in systems modeled by quadratic equations.
Economics and Finance
- Calculating break-even points where profit functions are quadratic.
- Analyzing quadratic cost or revenue functions to find maximum or minimum points.
Computer Graphics and Design
- Detecting intersections between curves and surfaces.
- Rendering shapes accurately by solving quadratic equations.
Summary: How to Find the Discriminant of 2x² + x - 6 = 0
To summarize, here's a quick step-by-step guide:
- Write the quadratic in standard form: \( ax^2 + bx + c = 0 \).
- Identify coefficients:
- \( a = 2 \),
- \( b = 1 \),
- \( c = -6 \).
\[ D = b^2 - 4ac \]
- Substitute the values:
\[ D = 1^2 - 4 \times 2 \times (-6) = 1 + 48 = 49 \]
- Interpret the discriminant:
- Since \( D > 0 \), the equation has two real roots.
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Conclusion
Calculating the discriminant of quadratic equations like 2x² + x - 6 = 0 is a straightforward yet powerful technique in algebra. It provides essential insights into the nature of the roots, saving time and effort when solving quadratic equations. Remember, the key steps involve identifying coefficients, applying the discriminant formula, and interpreting the result effectively.
Understanding the discriminant equips students and professionals with the ability to analyze quadratic equations quickly and accurately, which is invaluable in both academic settings and real-world applications. Practice regularly with different equations to strengthen your skills and deepen your understanding of this fundamental mathematical concept.
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Keywords: Discriminant, quadratic equation, 2x² + x - 6, roots, real solutions, quadratic formula, algebra, discriminant calculation, quadratic roots, properties of quadratic equations, solving quadratic equations.
