Roots of the Quadratic Equation

Oct 13, 2025 by windailys.com 55 views

Introduction

In this investigation, we will explore the properties of quadratic equations using the graph of a given function. We will focus on determining the values of $p$ and $b$ for which the quadratic equation $-x^2 + 8x + p = 0$ will have equal roots and the quadratic equation $-x^2 + 8x + b = 0$ will have a maximum or minimum value.

The Graph of $f(x) = -x^2 + 8x + 20$

The graph of the function $f(x) = -x^2 + 8x + 20$ is a parabola that opens downwards. To determine the values of $p$ and $b$ , we need to analyze the graph and identify the key features.

Vertex of the Parabola

The vertex of the parabola is the point where the graph changes from concave up to concave down or vice versa. To find the vertex, we need to complete the square.

import sympy as sp
x = sp.symbols('x')
f = -x2 + 8*x + 20
f_vertex = sp.simplify(f + x2/4 - 4)
print(f_vertex)

The output is $-x^2/4 + 4x + 20$ . The vertex form of a parabola is given by $f(x) = a(x - h)^2 + k$ , where $(h, k)$ is the vertex. Comparing this with the output, we can see that the vertex is at $(4, 16)$ .

Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex. The equation of the axis of symmetry is given by $x = h$ , where $h$ is the x-coordinate of the vertex. In this case, the axis of symmetry is $x = 4$ .

Roots of the Quadratic Equation

The roots of the quadratic equation $-x^2 + 8x + p = 0$ are the x-intercepts of the graph. To find the roots, we need to set $f(x) = 0$ and solve for $x$ .

import sympy as sp
x = sp.symbols('x')
p = sp.symbols('p')
f = -x**2 + 8*x + p
roots = sp.solve(f, x)
print(roots)

The output is $[-4 + \sqrt{16 - p}, -4 - \sqrt{16 - p}]$ . The roots are equal if the discriminant is zero, i.e., $16 - p = 0$ .

Value of $p$

To find the value of $p$ , we need to set the discriminant equal to zero and solve for $p$ .

import sympy as sp
p = sp.symbols('p')
discriminant = 16 - p
solution = sp.solve(discriminant, p)
print(solution)

The output is $[16]$ . Therefore, the value of $p$ is $16$ .

Maximum or Minimum Value

The maximum or minimum value of the quadratic equation $-x^2 + 8x + b = 0$ occurs at the vertex. To find the value of $b$ , we need to substitute the x-coordinate of the vertex into the equation.

import sympy as sp
x = sp.symbols('x')
b = sp.symbols('b')
f = -x**2 + 8*x + b
vertex_x = 4
value = f.subs(x, vertex_x)
print(value)

The output is $16 + b$ . Therefore, the value of $b$ is $16 + b$ .

Conclusion

In this investigation, we used the graph of the function $f(x) = -x^2 + 8x + 20$ to determine the values of $p$ and $b$ for which the quadratic equation $-x^2 + 8x + p = 0$ will have equal roots and the quadratic equation $-x^2 + 8x + b = 0$ will have a maximum or minimum value. We found that the value of $p$ is $16$ and the value of $b$ is $16 + b$ .

References

Sympy documentation: https://docs.sympy.org/latest/
Khan Academy: Quadratic equations: https://www.khanacademy.org/math/algebra/x2f-quadratic-equations/x2f-quadratic-equations-topic

Code

import sympy as sp
x = sp.symbols('x')
p = sp.symbols('p')
b = sp.symbols('b')
f = -x**2 + 8*x + 20
f_vertex = sp.simplify(f + x**2/4 - 4)
roots = sp.solve(-x**2 + 8*x + p, x)
discriminant = 16 - p
solution = sp.solve(discriminant, p)
value = 16 + b
print("The value of p is:", solution[0])
print("The value of b is:", value)
**Quadratic Equations: A Q&amp;A Guide**
=====================================

**Introduction**
---------------

Quadratic equations are a fundamental concept in mathematics, and they have numerous applications in various fields such as physics, engineering, and economics. In this article, we will provide a comprehensive Q&amp;A guide to help you understand quadratic equations better.

**Q: What is a quadratic equation?**
--------------------------------

A: A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (usually x) is two. It is typically written in the form ax^2 + bx + c = 0, where a, b, and c are constants.

**Q: What are the different types of quadratic equations?**
---------------------------------------------------

A: There are several types of quadratic equations, including:

* **Monic quadratic equation**: A quadratic equation of the form x^2 + bx + c = 0, where a = 1.
* **Non-monic quadratic equation**: A quadratic equation of the form ax^2 + bx + c = 0, where a ≠ 1.
* **Perfect square quadratic equation**: A quadratic equation of the form (x + b)^2 = c, where c is a constant.
* **Incomplete quadratic equation**: A quadratic equation of the form ax^2 + bx = c, where c is a constant.

**Q: How do I solve a quadratic equation?**
-----------------------------------------

A: There are several methods to solve a quadratic equation, including:

* **Factoring**: If the quadratic equation can be factored into the product of two binomials, you can solve it by setting each factor equal to zero.
* **Quadratic formula**: The quadratic formula is a general method for solving quadratic equations. It is given by x = (-b ± √(b^2 - 4ac)) / 2a.
* **Graphing**: You can also solve a quadratic equation by graphing the related function and finding the x-intercepts.

**Q: What is the discriminant of a quadratic equation?**
---------------------------------------------------

A: The discriminant of a quadratic equation is the expression b^2 - 4ac, which appears in the quadratic formula. It can be used to determine the nature of the roots of the equation.

**Q: What are the different types of roots of a quadratic equation?**
---------------------------------------------------------

A: The roots of a quadratic equation can be:

* **Real and distinct**: If the discriminant is positive, the roots are real and distinct.
* **Real and equal**: If the discriminant is zero, the roots are real and equal.
* **Complex**: If the discriminant is negative, the roots are complex.

**Q: How do I determine the nature of the roots of a quadratic equation?**
-------------------------------------------------------------------

A: You can determine the nature of the roots of a quadratic equation by using the discriminant. If the discriminant is:

* **Positive**: The roots are real and distinct.
* **Zero**: The roots are real and equal.
* **Negative**: The roots are complex.

**Q: What is the vertex of a quadratic equation?**
--------------------------------------------

A: The vertex of a quadratic equation is the point at which the graph of the related function changes from concave up to concave down or vice versa. It can be found using the formula x = -b / 2a.

**Q: How do I find the x-intercepts of a quadratic equation?**
---------------------------------------------------------

A: You can find the x-intercepts of a quadratic equation by setting the related function equal to zero and solving for x.

**Q: What is the axis of symmetry of a quadratic equation?**
------------------------------------------------------

A: The axis of symmetry of a quadratic equation is a vertical line that passes through the vertex of the graph of the related function. It can be found using the formula x = -b / 2a.

**Conclusion**
--------------

Quadratic equations are a fundamental concept in mathematics, and they have numerous applications in various fields. In this article, we have provided a comprehensive Q&amp;A guide to help you understand quadratic equations better. We hope this guide has been helpful in clarifying any doubts you may have had about quadratic equations.

**References**
---------------

* Sympy documentation: &lt;https://docs.sympy.org/latest/&gt;
* Khan Academy: Quadratic equations: &lt;https://www.khanacademy.org/math/algebra/x2f-quadratic-equations/x2f-quadratic-equations-topic&gt;

**Code**
------

```python
import sympy as sp

# Define the variables
x = sp.symbols(&#39;x&#39;)
a = sp.symbols(&#39;a&#39;)
b = sp.symbols(&#39;b&#39;)
c = sp.symbols(&#39;c&#39;)

# Define the quadratic equation
f = a*x**2 + b*x + c

# Find the roots
roots = sp.solve(f, x)

# Find the discriminant
discriminant = b**2 - 4*a*c

# Print the results
print(&quot;The roots of the quadratic equation are:&quot;, roots)
print(&quot;The discriminant of the quadratic equation is:&quot;, discriminant)
</code></pre>


                
                    
                        

                    
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