The First Quadratic Equation

Oct 13, 2025 by miodeya.com 5 views

Introduction

Quadratic equations are a fundamental concept in mathematics, and they play a crucial role in various fields, including algebra, geometry, and calculus. In this article, we will delve into two quadratic equations, $x^2 - 7x + 12 = 0$ and $3x^2 - 21x + 36 = 0$ , and explore why they have the same solutions despite their seemingly different forms.

The First Quadratic Equation

The first quadratic equation is $x^2 - 7x + 12 = 0$ . To factor this equation, we need to find two numbers whose product is $12$ and whose sum is $-7$ . These numbers are $-3$ and $-4$ , as their product is $12$ and their sum is $-7$ . Therefore, we can rewrite the equation as:

x^2 - 7x + 12 = (x - 3)(x - 4) = 0

This means that either $(x - 3) = 0$ or $(x - 4) = 0$ . Solving for $x$ , we get:

x - 3 = 0 \Rightarrow x = 3

x - 4 = 0 \Rightarrow x = 4

Therefore, the solutions to the first quadratic equation are $x = 3$ and $x = 4$ .

The Second Quadratic Equation

The second quadratic equation is $3x^2 - 21x + 36 = 0$ . To factor this equation, we need to find two numbers whose product is $36$ and whose sum is $-21$ . These numbers are $-9$ and $-4$ , as their product is $36$ and their sum is $-21$ . Therefore, we can rewrite the equation as:

3x^2 - 21x + 36 = 3(x^2 - 7x + 12) = 3(x - 3)(x - 4) = 0

This means that either $(x - 3) = 0$ or $(x - 4) = 0$ . Solving for $x$ , we get:

x - 3 = 0 \Rightarrow x = 3

x - 4 = 0 \Rightarrow x = 4

Therefore, the solutions to the second quadratic equation are also $x = 3$ and $x = 4$ .

Why Do They Have the Same Solutions?

The reason why the two quadratic equations have the same solutions is that they are equivalent. The second equation can be rewritten as $3(x^2 - 7x + 12) = 0$ , which is equivalent to the first equation $(x^2 - 7x + 12) = 0$ . This means that the two equations have the same roots, and therefore, they have the same solutions.

Conclusion

In conclusion, the two quadratic equations $x^2 - 7x + 12 = 0$ and $3x^2 - 21x + 36 = 0$ have the same solutions because they are equivalent. The second equation can be rewritten as $3(x^2 - 7x + 12) = 0$ , which is equivalent to the first equation $(x^2 - 7x + 12) = 0$ . This means that the two equations have the same roots, and therefore, they have the same solutions.

Key Takeaways

The two quadratic equations $x^2 - 7x + 12 = 0$ and $3x^2 - 21x + 36 = 0$ have the same solutions.
The second equation can be rewritten as $3(x^2 - 7x + 12) = 0$ , which is equivalent to the first equation $(x^2 - 7x + 12) = 0$ .
The two equations have the same roots, and therefore, they have the same solutions.

Frequently Asked Questions

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 often written in the form ax^2 + bx + c = 0, where a, b, and c are constants.

Q: How do I solve a quadratic equation?

A: There are several methods to solve a quadratic equation, including factoring, using the quadratic formula, and completing the square. The method you choose will depend on the specific equation and your personal preference.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that can be used to solve quadratic equations. It is written as:

x = (-b ± √(b^2 - 4ac)) / 2a

where a, b, and c are the coefficients of the quadratic equation.

Q: What is the difference between a quadratic equation and a linear equation?

A: A linear equation is a polynomial equation of degree one, which means the highest power of the variable (usually x) is one. A quadratic equation, on the other hand, is a polynomial equation of degree two, which means the highest power of the variable (usually x) is two.

Q: Can a quadratic equation have more than two solutions?

A: No, a quadratic equation can only have two solutions. This is because the graph of a quadratic equation is a parabola, which has a maximum or minimum point, but no more than two x-intercepts.

Q: How do I determine the number of solutions a quadratic equation has?

A: To determine the number of solutions a quadratic equation has, you can use the discriminant, which is the expression b^2 - 4ac. If the discriminant is positive, the equation has two distinct solutions. If the discriminant is zero, the equation has one repeated solution. If the discriminant is negative, the equation has no real solutions.

Q: What is the discriminant?

A: The discriminant is the expression b^2 - 4ac, where a, b, and c are the coefficients of the quadratic equation. It is used to determine the number of solutions a quadratic equation has.

Q: Can a quadratic equation have complex solutions?

A: Yes, a quadratic equation can have complex solutions. If the discriminant is negative, the equation has no real solutions, but it may have complex solutions.

Q: How do I find the complex solutions of a quadratic equation?

A: To find the complex solutions of a quadratic equation, you can use the quadratic formula and take the square root of the negative discriminant. This will give you two complex solutions.

Q: What is the relationship between the solutions of a quadratic equation and its graph?

A: The solutions of a quadratic equation are the x-intercepts of its graph. The graph of a quadratic equation is a parabola, which has a maximum or minimum point, but no more than two x-intercepts.

Q: Can a quadratic equation have a vertical asymptote?

A: No, a quadratic equation cannot have a vertical asymptote. This is because the graph of a quadratic equation is a parabola, which has a maximum or minimum point, but no vertical asymptotes.

Q: How do I graph a quadratic equation?

A: To graph a quadratic equation, you can use a graphing calculator or a computer program. You can also use a table of values to plot the points on the graph and then draw the parabola.

Q: What is the significance of the vertex of a quadratic equation?

A: The vertex of a quadratic equation is the maximum or minimum point of its graph. It is the point where the parabola changes direction, and it is an important point in the graph of a quadratic equation.

Q: Can a quadratic equation have a horizontal asymptote?

A: No, a quadratic equation cannot have a horizontal asymptote. This is because the graph of a quadratic equation is a parabola, which has a maximum or minimum point, but no horizontal asymptotes.

Q: How do I find the vertex of a quadratic equation?

A: To find the vertex of a quadratic equation, you can use the formula x = -b / 2a, where a, b, and c are the coefficients of the quadratic equation. This will give you the x-coordinate of the vertex. To find the y-coordinate of the vertex, you can plug the x-coordinate into the equation and solve for y.

Q: What is the relationship between the vertex of a quadratic equation and its solutions?

A: The vertex of a quadratic equation is the midpoint of its solutions. This means that the x-coordinate of the vertex is the average of the x-coordinates of the solutions.

Q: Can a quadratic equation have a slant asymptote?

A: No, a quadratic equation cannot have a slant asymptote. This is because the graph of a quadratic equation is a parabola, which has a maximum or minimum point, but no slant asymptotes.

Q: How do I find the slant asymptote of a quadratic equation?

A: A quadratic equation cannot have a slant asymptote. This is because the graph of a quadratic equation is a parabola, which has a maximum or minimum point, but no slant asymptotes.

Conclusion

In conclusion, quadratic equations are an important concept in mathematics, and they have many applications in various fields. By understanding the properties and solutions of quadratic equations, you can solve a wide range of problems and make informed decisions.