What Are The Two Solutions Of $x^2-2x-4=-3x+9$?A. The Y Y Y -coordinates Of The Y Y Y -intercepts Of The Graphs Of Y = X 2 − 2 X − 4 Y=x^2-2x-4 Y = X 2 − 2 X − 4 And Y = − 3 X + 9 Y=-3x+9 Y = − 3 X + 9 .B. The X X X -coordinates Of The X X X -intercepts Of The

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Introduction

Solving quadratic equations is a fundamental concept in mathematics, and it has numerous applications in various fields such as physics, engineering, and economics. In this article, we will explore the solutions of the quadratic equation $x^2-2x-4=-3x+9$ and provide a detailed explanation of the two solutions.

Understanding the Equation

The given equation is a quadratic equation in the form of $ax^2+bx+c=0$, where $a=1$, $b=-2$, and $c=-4$. To solve this equation, we need to isolate the variable $x$ and find its values that satisfy the equation.

Rearranging the Equation

To solve the equation, we first need to rearrange it to the standard form of a quadratic equation. We can do this by moving all the terms to one side of the equation:

x2−2x−4+3x−9=0x^2-2x-4+3x-9=0

Simplifying the equation, we get:

x2+x−13=0x^2+x-13=0

Factoring the Equation

One way to solve the quadratic equation is to factor it. However, in this case, the equation does not factor easily. Therefore, we will use the quadratic formula to find the solutions.

Quadratic Formula

The quadratic formula is given by:

x=−b±b2−4ac2ax=\frac{-b\pm\sqrt{b^2-4ac}}{2a}

In this case, $a=1$, $b=1$, and $c=-13$. Plugging these values into the formula, we get:

x=−1±12−4(1)(−13)2(1)x=\frac{-1\pm\sqrt{1^2-4(1)(-13)}}{2(1)}

Simplifying the expression under the square root, we get:

x=−1±1+522x=\frac{-1\pm\sqrt{1+52}}{2}

x=−1±532x=\frac{-1\pm\sqrt{53}}{2}

Solutions of the Equation

The two solutions of the equation are given by:

x1=−1+532x_1=\frac{-1+\sqrt{53}}{2}

x2=−1−532x_2=\frac{-1-\sqrt{53}}{2}

Explanation of the Solutions

The two solutions of the equation represent the x-coordinates of the x-intercepts of the graphs of the two functions:

y=x2+x−13y=x^2+x-13

y=−3x+9y=-3x+9

The x-intercepts of a function are the points where the function intersects the x-axis. In this case, the two functions intersect the x-axis at the points $(x_1,0)$ and $(x_2,0)$.

Conclusion

In conclusion, the two solutions of the quadratic equation $x^2-2x-4=-3x+9$ are given by:

x1=−1+532x_1=\frac{-1+\sqrt{53}}{2}

x2=−1−532x_2=\frac{-1-\sqrt{53}}{2}

These solutions represent the x-coordinates of the x-intercepts of the graphs of the two functions:

y=x2+x−13y=x^2+x-13

y=−3x+9y=-3x+9

Final Answer

The final answer is: −1+532,−1−532\boxed{\frac{-1+\sqrt{53}}{2},\frac{-1-\sqrt{53}}{2}}

Introduction

In our previous article, we explored the solutions of the quadratic equation $x^2-2x-4=-3x+9$. We found that the two solutions of the equation are given by:

x1=−1+532x_1=\frac{-1+\sqrt{53}}{2}

x2=−1−532x_2=\frac{-1-\sqrt{53}}{2}

In this article, we will answer some frequently asked questions about the solutions of the quadratic equation.

Q: What are the x-coordinates of the x-intercepts of the graphs of the two functions?

A: The x-coordinates of the x-intercepts of the graphs of the two functions are given by the two solutions of the quadratic equation:

x1=−1+532x_1=\frac{-1+\sqrt{53}}{2}

x2=−1−532x_2=\frac{-1-\sqrt{53}}{2}

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

A: To find the solutions of a quadratic equation, you can use the quadratic formula:

x=−b±b2−4ac2ax=\frac{-b\pm\sqrt{b^2-4ac}}{2a}

You can also try to factor the equation, but in some cases, the equation may not factor easily.

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

A: The solutions of a quadratic equation represent the x-coordinates of the x-intercepts of the graph of the function. In other words, they represent the points where the function intersects the x-axis.

Q: Can you explain the concept of x-intercepts?

A: Yes, the x-intercepts of a function are the points where the function intersects the x-axis. In other words, they are the points where the function has a value of zero.

Q: How do you graph a quadratic function?

A: To graph a quadratic function, you can start by finding the x-intercepts of the function. You can then use these points to draw the graph of the function.

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

A: The solutions of a quadratic equation represent the x-coordinates of the x-intercepts of the graph of the function. In other words, they represent the points where the function intersects the x-axis.

Q: Can you provide an example of a quadratic equation and its solutions?

A: Yes, consider the quadratic equation $x^2-2x-4=-3x+9$. The solutions of this equation are given by:

x1=−1+532x_1=\frac{-1+\sqrt{53}}{2}

x2=−1−532x_2=\frac{-1-\sqrt{53}}{2}

These solutions represent the x-coordinates of the x-intercepts of the graph of the function.

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

A: To determine the number of solutions of a quadratic equation, you can use the discriminant:

b2−4acb^2-4ac

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 real solutions.

Q: Can you explain the concept of the discriminant?

A: Yes, the discriminant is a value that is used to determine the number of solutions of a quadratic equation. It is given by:

b2−4acb^2-4ac

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 real solutions.

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

A: To use the discriminant to determine the number of solutions of a quadratic equation, you can follow these steps:

  1. Calculate the discriminant:

b2−4acb^2-4ac

  1. If the discriminant is positive, the equation has two solutions.
  2. If the discriminant is zero, the equation has one solution.
  3. If the discriminant is negative, the equation has no real solutions.

Conclusion

In conclusion, the solutions of a quadratic equation represent the x-coordinates of the x-intercepts of the graph of the function. The quadratic formula can be used to find the solutions of a quadratic equation. The discriminant can be used to determine the number of solutions of a quadratic equation.

Final Answer

The final answer is: −1+532,−1−532\boxed{\frac{-1+\sqrt{53}}{2},\frac{-1-\sqrt{53}}{2}}