Consider The Graph Of The Quadratic Function $y = 3x^2 - 3x - 6$.What Are The Solutions Of The Quadratic Equation $0 = 3x^2 - 3x - 6$?A. -1 And 2 B. -6 And -1 C. -1 And 1 D. -6 And 2

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 this article, we will focus on solving quadratic equations, specifically the equation $0 = 3x^2 - 3x - 6$. We will use the graph of the quadratic function $y = 3x^2 - 3x - 6$ to find the solutions of the quadratic equation.

Understanding Quadratic Equations

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants. In our case, the quadratic equation is $0 = 3x^2 - 3x - 6$.

Graphing the Quadratic Function

To find the solutions of the quadratic equation, we need to graph the quadratic function $y = 3x^2 - 3x - 6$. The graph of a quadratic function is a parabola, which is a U-shaped curve. The parabola opens upward if $a > 0$ and downward if $a < 0$. In our case, $a = 3 > 0$, so the parabola opens upward.

Finding the Vertex

The vertex of a parabola is the lowest or highest point on the curve. To find the vertex, we need to find the x-coordinate of the vertex, which is given by the formula $x = -\frac{b}{2a}$. In our case, $a = 3$ and $b = -3$, so the x-coordinate of the vertex is $x = -\frac{-3}{2(3)} = \frac{1}{2}$.

Finding the y-Coordinate of the Vertex

To find the y-coordinate of the vertex, we need to plug the x-coordinate into the equation of the quadratic function. In our case, the x-coordinate of the vertex is $x = \frac{1}{2}$, so the y-coordinate of the vertex is $y = 3(\frac{1}{2})^2 - 3(\frac{1}{2}) - 6 = -\frac{27}{4}$.

Finding the Solutions

The solutions of the quadratic equation are the x-coordinates of the points where the parabola intersects the x-axis. To find the solutions, we need to set the equation of the quadratic function equal to zero and solve for x. In our case, the equation is $0 = 3x^2 - 3x - 6$. We can factor the left-hand side of the equation as $(3x + 2)(x - 3) = 0$.

Factoring the Quadratic Expression

To factor the quadratic expression $(3x + 2)(x - 3)$, we need to find two numbers whose product is $-6$ and whose sum is $-3$. The two numbers are $-6$ and $2$, so we can write the quadratic expression as $(3x + 2)(x - 3) = (3x + 2)(x - 3) = 0$.

Solving for x

To solve for x, we need to set each factor equal to zero and solve for x. In our case, we have two factors: $3x + 2 = 0$ and $x - 3 = 0$. We can solve each factor separately.

Solving the First Factor

To solve the first factor $3x + 2 = 0$, we need to isolate x. We can do this by subtracting 2 from both sides of the equation and then dividing both sides by 3. This gives us $x = -\frac{2}{3}$.

Solving the Second Factor

To solve the second factor $x - 3 = 0$, we need to isolate x. We can do this by adding 3 to both sides of the equation. This gives us $x = 3$.

Conclusion

In this article, we have solved the quadratic equation $0 = 3x^2 - 3x - 6$ using the graph of the quadratic function $y = 3x^2 - 3x - 6$. We have found the solutions of the quadratic equation to be $x = -\frac{2}{3}$ and $x = 3$. These solutions are the x-coordinates of the points where the parabola intersects the x-axis.

Final Answer

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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 solved the quadratic equation $0 = 3x^2 - 3x - 6$ using the graph of the quadratic function $y = 3x^2 - 3x - 6$. 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. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants.

Q: How do I graph a quadratic function?

A: To graph a quadratic function, you need to find the vertex of the parabola. The vertex is the lowest or highest point on the curve. You can find the vertex by using the formula $x = -\frac{b}{2a}$, where $a$ and $b$ are the coefficients of the quadratic function.

Q: What is the vertex of a parabola?

A: The vertex of a parabola is the lowest or highest point on the curve. It is the point where the parabola changes direction.

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

A: To find the solutions of a quadratic equation, you need to set the equation equal to zero and solve for x. You can use factoring, the quadratic formula, or graphing to find the solutions.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that gives the solutions of a quadratic equation. It is given by the formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a$, $b$, and $c$ are the coefficients of the quadratic equation.

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. You will then get two solutions for x.

Q: What are the different types of quadratic equations?

A: There are three types of quadratic equations: monic, non-monic, and quadratic equations with complex roots.

Q: What is a monic quadratic equation?

A: A monic quadratic equation is a quadratic equation of the form $x^2 + bx + c = 0$, where $a = 1$.

Q: What is a non-monic quadratic equation?

A: A non-monic quadratic equation is a quadratic equation of the form $ax^2 + bx + c = 0$, where $a \neq 1$.

Q: What is a quadratic equation with complex roots?

A: A quadratic equation with complex roots is a quadratic equation that has complex solutions.

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

A: To determine the number of solutions of a quadratic equation, you need to look at the discriminant, which is given by the formula $b^2 - 4ac$. 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.

Conclusion

In this article, we have provided a Q&A guide to help you understand quadratic equations better. We have covered topics such as graphing quadratic functions, finding the solutions of quadratic equations, and determining the number of solutions of a quadratic equation.

Final Answer

The final answer is: $\boxed{0}$