If $(x-1)(x-2) = 3x - 7$, How Many Unique Values Of $x$ Exist?

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Introduction

In this article, we will explore the concept of solving quadratic equations and determine the number of unique values of $x$ that satisfy the given equation. The equation $(x-1)(x-2) = 3x - 7$ is a quadratic equation, and we will use various algebraic techniques to solve it.

Understanding the Equation

The given equation is a quadratic equation in the form of $(x-a)(x-b) = c$, where $a$, $b$, and $c$ are constants. To solve this equation, we need to expand the left-hand side and then simplify the resulting expression.

Expanding the Left-Hand Side

To expand the left-hand side of the equation, we need to multiply the two binomials using the distributive property. This gives us:

(xβˆ’1)(xβˆ’2)=x2βˆ’2xβˆ’x+2(x-1)(x-2) = x^2 - 2x - x + 2

Simplifying the expression, we get:

x2βˆ’3x+2x^2 - 3x + 2

Equating the Expressions

Now that we have expanded the left-hand side of the equation, we can equate it to the right-hand side:

x2βˆ’3x+2=3xβˆ’7x^2 - 3x + 2 = 3x - 7

Simplifying the Equation

To simplify the equation, we need to move all the terms to one side of the equation. This gives us:

x2βˆ’3x+2βˆ’3x+7=0x^2 - 3x + 2 - 3x + 7 = 0

Combining like terms, we get:

x2βˆ’6x+9=0x^2 - 6x + 9 = 0

Solving the Quadratic Equation

The equation $x^2 - 6x + 9 = 0$ is a quadratic equation in the form of $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants. To solve this equation, we can use the quadratic formula:

x=βˆ’bΒ±b2βˆ’4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

In this case, $a = 1$, $b = -6$, and $c = 9$. Plugging these values into the quadratic formula, we get:

x=βˆ’(βˆ’6)Β±(βˆ’6)2βˆ’4(1)(9)2(1)x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(1)(9)}}{2(1)}

Simplifying the expression, we get:

x=6Β±36βˆ’362x = \frac{6 \pm \sqrt{36 - 36}}{2}

x=6Β±02x = \frac{6 \pm \sqrt{0}}{2}

x=62x = \frac{6}{2}

x=3x = 3

Conclusion

In this article, we have solved the quadratic equation $(x-1)(x-2) = 3x - 7$ and found that there is only one unique value of $x$ that satisfies the equation. The value of $x$ is $3$.

Final Thoughts

The concept of solving quadratic equations is an important one in mathematics, and it has many real-world applications. In this article, we have used various algebraic techniques to solve a quadratic equation and found the number of unique values of $x$ that satisfy the equation. We hope that this article has provided a clear and concise explanation of the concept of solving quadratic equations and has helped readers to understand the material better.

Additional Resources

For more information on solving quadratic equations, we recommend the following resources:

  • Khan Academy: Quadratic Equations
  • Mathway: Quadratic Equation Solver
  • Wolfram Alpha: Quadratic Equation Solver

Frequently Asked Questions

Q: What is a quadratic equation? A: A quadratic equation is a polynomial equation of degree two, which means that the highest power of the variable is two.

Q: How do I solve a quadratic equation? A: To solve a quadratic equation, you can use the quadratic formula or factor the equation.

Q: What is the quadratic formula? A: The quadratic formula is a formula that is used to solve quadratic equations. It is given by:

x=βˆ’bΒ±b2βˆ’4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Q: How do I factor a quadratic equation? A: To factor a quadratic equation, you need to find two numbers whose product is equal to the constant term and whose sum is equal to the coefficient of the linear term.

Conclusion

In conclusion, solving quadratic equations is an important concept in mathematics, and it has many real-world applications. In this article, we have used various algebraic techniques to solve a quadratic equation and found the number of unique values of $x$ that satisfy the equation. We hope that this article has provided a clear and concise explanation of the concept of solving quadratic equations and has helped readers to understand the material better.

Introduction

In our previous article, we explored the concept of solving quadratic equations and determined the number of unique values of $x$ that satisfy the given equation. In this article, we will answer some frequently asked questions about quadratic equations and provide additional resources for further learning.

Q&A

Q: What is a quadratic equation?

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

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you can use the quadratic formula or factor the equation. The quadratic formula is given by:

x=βˆ’bΒ±b2βˆ’4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Q: What is the quadratic formula?

A: The quadratic formula is a formula that is used to solve quadratic equations. It is given by:

x=βˆ’bΒ±b2βˆ’4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Q: How do I factor a quadratic equation?

A: To factor a quadratic equation, you need to find two numbers whose product is equal to the constant term and whose sum is equal to the coefficient of the linear term.

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 highest power of two, while a linear equation has a highest power of one.

Q: Can I use the quadratic formula to solve a linear equation?

A: No, you cannot use the quadratic formula to solve a linear equation. The quadratic formula is used to solve quadratic equations, while a linear equation can be solved using other methods, such as substitution or elimination.

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

A: To determine the number of solutions to a quadratic equation, you need to examine the discriminant, which is given by $b^2 - 4ac$. 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 a value that is used to determine the number of solutions to a quadratic equation. It is given by $b^2 - 4ac$.

Q: Can I use the quadratic formula to solve a quadratic equation with complex solutions?

A: Yes, you can use the quadratic formula to solve a quadratic equation with complex solutions. The quadratic formula will give you two complex solutions.

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 graph.

Q: What is the vertex of a quadratic equation?

A: The vertex of a quadratic equation is the point on the graph where the parabola changes direction. It is given by the formula $x = -\frac{b}{2a}$.

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 $x = -\frac{b}{2a}$. You can then substitute this value into the equation to find the corresponding y-coordinate.

Conclusion

In this article, we have answered some frequently asked questions about quadratic equations and provided additional resources for further learning. We hope that this article has provided a clear and concise explanation of the concept of quadratic equations and has helped readers to understand the material better.

Additional Resources

For more information on quadratic equations, we recommend the following resources:

  • Khan Academy: Quadratic Equations
  • Mathway: Quadratic Equation Solver
  • Wolfram Alpha: Quadratic Equation Solver
  • MIT OpenCourseWare: Quadratic Equations
  • Wolfram MathWorld: Quadratic Equations

Frequently Asked Questions

Q: What is a quadratic equation? A: A quadratic equation is a polynomial equation of degree two, which means that the highest power of the variable is two.

Q: How do I solve a quadratic equation? A: To solve a quadratic equation, you can use the quadratic formula or factor the equation.

Q: What is the quadratic formula? A: The quadratic formula is a formula that is used to solve quadratic equations. It is given by:

x=βˆ’bΒ±b2βˆ’4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Q: How do I factor a quadratic equation? A: To factor a quadratic equation, you need to find two numbers whose product is equal to the constant term and whose sum is equal to the coefficient of the linear term.

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.

Q: Can I use the quadratic formula to solve a linear equation? A: No, you cannot use the quadratic formula to solve a linear equation. The quadratic formula is used to solve quadratic equations, while a linear equation can be solved using other methods, such as substitution or elimination.

Q: How do I determine the number of solutions to a quadratic equation? A: To determine the number of solutions to a quadratic equation, you need to examine the discriminant, which is given by $b^2 - 4ac$. 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 a value that is used to determine the number of solutions to a quadratic equation. It is given by $b^2 - 4ac$.

Q: Can I use the quadratic formula to solve a quadratic equation with complex solutions? A: Yes, you can use the quadratic formula to solve a quadratic equation with complex solutions. The quadratic formula will give you two complex solutions.

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 graph.

Q: What is the vertex of a quadratic equation? A: The vertex of a quadratic equation is the point on the graph where the parabola changes direction. It is given by the formula $x = -\frac{b}{2a}$.

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 $x = -\frac{b}{2a}$. You can then substitute this value into the equation to find the corresponding y-coordinate.