Find The Number Of Real Number Solutions For The Equation:$x^2 - 2x + 9 = 0$A. 0 B. 2 C. 1

Introduction

Quadratic equations are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and economics. A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. In this article, we will focus on finding the number of real number solutions for the quadratic equation $x^2 - 2x + 9 = 0$. We will use various methods to solve this equation and determine the number of real solutions.

Understanding Quadratic Equations

A quadratic equation is generally written in the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $x$ is the variable. The graph of a quadratic equation is a parabola, which is a U-shaped curve. The parabola can open upwards or downwards, depending on the value of $a$. If $a$ is positive, the parabola opens upwards, and if $a$ is negative, the parabola opens downwards.

The Quadratic Formula

One of the most common methods for solving quadratic equations is the quadratic formula. The quadratic formula is given by:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

This formula gives two solutions for the quadratic equation, which are the x-coordinates of the points where the parabola intersects the x-axis.

Applying the Quadratic Formula

Now, let's apply the quadratic formula to the equation $x^2 - 2x + 9 = 0$. We have $a = 1$, $b = -2$, and $c = 9$. Plugging these values into the quadratic formula, we get:

$x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(9)}}{2(1)}$

Simplifying the expression, we get:

$x = \frac{2 \pm \sqrt{4 - 36}}{2}$

$x = \frac{2 \pm \sqrt{-32}}{2}$

Complex Solutions

The expression $\sqrt{-32}$ is an imaginary number, which means that the solutions to the equation are complex numbers. Complex numbers are numbers that have both real and imaginary parts. In this case, the solutions are:

$x = \frac{2 \pm 4i\sqrt{2}}{2}$

$x = 1 \pm 2i\sqrt{2}$

Conclusion

In conclusion, the quadratic equation $x^2 - 2x + 9 = 0$ has no real number solutions. The solutions to the equation are complex numbers, which means that they have both real and imaginary parts. This is because the discriminant $b^2 - 4ac$ is negative, which means that the parabola does not intersect the x-axis.

The Number of Real Solutions

The number of real solutions to a quadratic equation is determined by the discriminant. If the discriminant is positive, the equation has two real solutions. If the discriminant is zero, the equation has one real solution. If the discriminant is negative, the equation has no real solutions.

Real-World Applications

Quadratic equations have many real-world applications. For example, they are used to model the motion of objects under the influence of gravity. They are also used to determine the maximum or minimum value of a function.

Final Thoughts

In this article, we have seen how to find the number of real number solutions for a quadratic equation. We have used the quadratic formula to solve the equation and determined that it has no real solutions. We have also discussed the real-world applications of quadratic equations and the importance of understanding them.

References

• [1] "Quadratic Equations" by Math Open Reference
• [2] "Quadratic Formula" by Khan Academy
• [3] "Complex Numbers" by Wolfram MathWorld

Glossary

• Quadratic Equation: A polynomial equation of degree two, which means the highest power of the variable is two.
• Discriminant: The expression $b^2 - 4ac$ in the quadratic formula, which determines the number of real solutions to the equation.
• Complex Number: A number that has both real and imaginary parts.
• Parabola: The graph of a quadratic equation, which is a U-shaped curve.
  

Introduction

Quadratic equations are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and economics. In our previous article, we discussed how to find the number of real number solutions for a quadratic equation. In this article, we will provide a Q&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 is two. It is generally written in the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $x$ is the variable.

Q: What is the quadratic formula?

A: The quadratic formula is a method for solving quadratic equations. It is given by:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

This formula gives two solutions for the quadratic equation, which are the x-coordinates of the points where the parabola intersects the x-axis.

Q: What is the discriminant?

A: The discriminant is the expression $b^2 - 4ac$ in the quadratic formula. It determines the number of real solutions to the equation. If the discriminant is positive, the equation has two real solutions. If the discriminant is zero, the equation has one real solution. If the discriminant is negative, the equation has no real solutions.

Q: What is a complex number?

A: A complex number is a number that has both real and imaginary parts. It is generally written in the form $a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit.

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

A: To determine the number of real solutions to a quadratic equation, you need to calculate the discriminant. If the discriminant is positive, the equation has two real solutions. If the discriminant is zero, the equation has one real solution. If the discriminant is negative, the equation has no real solutions.

Q: What are some real-world applications of quadratic equations?

A: Quadratic equations have many real-world applications. For example, they are used to model the motion of objects under the influence of gravity. They are also used to determine the maximum or minimum value of a function.

Q: How do I solve a quadratic equation using the quadratic formula?

A: To solve a quadratic equation using the quadratic formula, you need to plug in the values of $a$, $b$, and $c$ into the formula. Then, simplify the expression and solve for $x$.

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

A: A quadratic equation is a polynomial equation of degree two, while a linear equation is a polynomial equation of degree one. A quadratic equation has a parabolic graph, while a linear equation has a straight line graph.

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

A: No, a quadratic equation can have at most two solutions. This is because the quadratic formula gives two solutions for the equation, and there are no other solutions.

Q: How do I graph a quadratic equation?

A: To graph a quadratic equation, you need to plot the points where the parabola intersects the x-axis. You can use the quadratic formula to find these points.

Q: What is the vertex of a parabola?

A: The vertex of a parabola is the point where the parabola changes direction. It is the minimum or maximum point of the parabola.

Q: How do I find the vertex of a parabola?

A: To find the vertex of a parabola, you need to use the formula $x = -\frac{b}{2a}$. This will give you the x-coordinate of the vertex. Then, you can plug this value into the equation to find the y-coordinate of the vertex.

Conclusion

In this article, we have provided a Q&A guide to help you understand quadratic equations better. We have discussed various topics related to quadratic equations, including the quadratic formula, discriminant, complex numbers, and real-world applications. We hope that this guide has been helpful in clarifying any doubts you may have had about quadratic equations.

References

• [1] "Quadratic Equations" by Math Open Reference
• [2] "Quadratic Formula" by Khan Academy
• [3] "Complex Numbers" by Wolfram MathWorld

Glossary

• Quadratic Equation: A polynomial equation of degree two, which means the highest power of the variable is two.
• Discriminant: The expression $b^2 - 4ac$ in the quadratic formula, which determines the number of real solutions to the equation.
• Complex Number: A number that has both real and imaginary parts.
• Parabola: The graph of a quadratic equation, which is a U-shaped curve.
• Vertex: The point where the parabola changes direction, which is the minimum or maximum point of the parabola.