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Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving a quadratic equation of the form $v^2 - 8v + 12 = 0$. We will use the quadratic formula to find the solutions and provide a step-by-step guide on how to solve it.

What is a Quadratic Equation?

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. It is typically written in the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $x$ is the variable. Quadratic equations can be solved using various methods, including factoring, completing the square, and the quadratic formula.

The Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It is given by:

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

where $a$, $b$, and $c$ are the coefficients of the quadratic equation. In our case, the equation is $v^2 - 8v + 12 = 0$, so we have:

$$a = 1, b = -8, c = 12$$

Solving the Quadratic Equation

Now that we have the coefficients, we can plug them into the quadratic formula:

$$v = \frac{-(-8) \pm \sqrt{(-8)^2 - 4(1)(12)}}{2(1)}$$

Simplifying the expression, we get:

$$v = \frac{8 \pm \sqrt{64 - 48}}{2}$$

$$v = \frac{8 \pm \sqrt{16}}{2}$$

$$v = \frac{8 \pm 4}{2}$$

This gives us two possible solutions:

$$v = \frac{8 + 4}{2} = 6$$

$$v = \frac{8 - 4}{2} = 2$$

Conclusion

In this article, we solved a quadratic equation of the form $v^2 - 8v + 12 = 0$ using the quadratic formula. We found two possible solutions: $v = 6$ and $v = 2$. This demonstrates the power of the quadratic formula in solving quadratic equations.

Discussion

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. The quadratic formula is a powerful tool for solving quadratic equations, and it can be used to find the solutions to a wide range of problems. In this article, we provided a step-by-step guide on how to solve a quadratic equation using the quadratic formula.

Additional Resources

For more information on solving quadratic equations, check out the following resources:

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

Final Answer

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Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them can be a challenging task for many students and professionals. In this article, we will provide a comprehensive Q&A guide on quadratic equations, covering frequently asked questions and providing detailed explanations.

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 typically written in the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $x$ is the variable.

Q: How do I solve a quadratic equation?

A: There are several methods to solve a quadratic equation, including factoring, completing the square, and the quadratic formula. The quadratic formula is a powerful tool for solving quadratic equations and is given by:

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

Q: What is the quadratic formula?

A: The quadratic formula is a formula used to solve quadratic equations of the form $ax^2 + bx + c = 0$. It is given by:

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

Q: How do I use the quadratic formula?

A: To use the quadratic formula, you need to plug in the values of $a$, $b$, and $c$ into the formula. For example, if you have the equation $x^2 + 4x + 4 = 0$, you would plug in $a = 1$, $b = 4$, and $c = 4$ into the formula.

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 a quadratic equation have more than one solution?

A: Yes, a quadratic equation can have more than one solution. In fact, a quadratic equation can have two solutions, one solution, or no solution at all.

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 look at the discriminant, which is the expression under the square root in the quadratic formula. If the discriminant is positive, the equation has two solutions. If the discriminant is zero, the equation has one solution. If the discriminant is negative, the equation has no solution.

Q: What is the discriminant?

A: The discriminant is the expression under the square root in the quadratic formula. It is given by:

$$b^2 - 4ac$$

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

A: To use the discriminant to determine the number of solutions to a quadratic equation, you need to plug in the values of $a$, $b$, and $c$ into the discriminant formula. If the result is positive, the equation has two solutions. If the result is zero, the equation has one solution. If the result is negative, the equation has no solution.

Q: Can a quadratic equation have a negative solution?

A: Yes, a quadratic equation can have a negative solution. In fact, a quadratic equation can have any type of solution, including positive, negative, or complex solutions.

Q: How do I graph a quadratic equation?

A: To graph a quadratic equation, you need to use a graphing calculator or a computer program. You can also use a graphing app on your phone or tablet.

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:

$$x = -\frac{b}{2a}$$

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

A: To find the vertex of a quadratic equation, you need to plug in the values of $a$ and $b$ into the vertex formula.