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Chapter 9: Problem 20

In the quadratic equation \(5 x^{2}-3=0, a=\) _____, \(b=\) _____, and \(c=\) _____.

Short Answer

Expert verified
In the given equation, \(a = 5\), \(b = 0\), and \(c = -3\).

Step by step solution

01

Identify the Form of a Quadratic Equation

The general form of a quadratic equation is \(ax^2 + bx + c = 0\). In this form, \(a\), \(b\), and \(c\) are the coefficients of the equation.
02

Comparison

The given equation is \(5x^2 - 3 = 0\). Comparing this to the general form, we see that \(a\) corresponds to the coefficient of \(x^2\), \(b\) corresponds to the coefficient of \(x\) and \(c\) is the constant term.
03

Determine the Coefficients

In \(5x^2 - 3 = 0\), the coefficient of \(x^2\) is 5, so \(a = 5\). There is no \(x\) term, so the coefficient \(b\) is 0, hence \(b = 0\). The constant term is -3, so \(c = -3\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Coefficients in Quadratic Equations
Coefficients are key components of quadratic equations. In a standard quadratic equation of the form \(ax^2 + bx + c = 0\), the letters \(a\), \(b\), and \(c\) represent the coefficients. Here's what each of these represents in the equation:
  • \(a\): This is the coefficient of \(x^2\), the quadratic term. It tells us how "steep" or "wide" the parabola will be.
  • \(b\): This is the coefficient of \(x\), the linear term. This term impacts the direction and position of the parabola.
  • \(c\): This is the constant term, representing the y-intercept of the parabola on the graph.
For the equation given: \(5x^2 - 3 = 0\):
  • \(a = 5\): This means the parabola will open upwards, and it will be fairly narrow, indicating a steep curve.
  • \(b = 0\): The absence of a linear term indicates the parabola is centered symmetrically about the y-axis.
  • \(c = -3\): This tells us that the parabola crosses the y-axis at \(y = -3\).
Understanding coefficients helps us predict the shape and position of the parabola without detailed graphing.
Using the Quadratic Formula
The quadratic formula is a powerful tool to find the roots of any quadratic equation. The formula is given by:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
This formula directly utilizes the coefficients from the quadratic equation \(ax^2 + bx + c = 0\):
  • \(-b\): The opposite of the linear coefficient.
  • \(b^2 - 4ac\): Known as the discriminant, this part of the formula tells us about the nature of the roots (real and distinct, real and repeated, or complex).
  • \(2a\): The denominator adjusts the influence of the quadratic term's steepness.
In our simple case, \(b = 0\), so the formula simplifies:
\[x = \sqrt{-\frac{c}{a}}\]This reduced form can quickly find roots when the middle term is missing.
Exploring Mathematical Concepts through Quadratics
Quadratic equations offer a beautiful intersection of algebra and geometry, allowing us to explore mathematical concepts such as functions, graphs, and complex numbers. Here's how some core concepts apply:
  • Functions: Quadratic equations represent parabolic functions, meaning each input \(x\) maps to one output \(y\).
  • Graphs: Graphing a quadratic equation gives us a parabola, a symmetric curve that gives insights into the nature of quadratic functions.
  • Symmetry: Many quadratic equations exhibit vertical symmetry about an axis, providing an aesthetic simplicity to their study.
  • Vertex: The highest or lowest point on the parabola is critical in real-world applications, from physics to economics.
  • Complex Numbers: When quadratic equations have no real roots, these introduce students to imaginary numbers and complex plane transformations.
By studying quadratics, learners gain insight into broader mathematical theory and practical applications, enhancing analytical skills and problem-solving abilities.

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Most popular questions from this chapter

Solve each equation. Write the answer in bi or a \(+\) bi form. $$2 x^{2}=-25$$

Use the quadratic formula to find all real solutions of each equation. SEE EXAMPLE \(2 .\) (OBJECTIVE 1) $$3 x^{2}+6=11 x$$

Solve each equation. Give both the exact answer and a decimal approximation to the nearest tenth. $$4 x^{2}+3 x-2=0$$

Use the quadratic formula to find all real solutions of each equation. SEE EXAMPLE 3. (OBJECTIVE 1) $$2 x^{2}-3=4 x$$

Solve each equation. Write the answer in bi or a \(+\) bi form. $$3 x^{2}-4 x+2=0$$
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