What Are The Two Solutions Of X 2 − 2 X − 4 = − 3 X + 9 X^2 - 2x - 4 = -3x + 9 X 2 − 2 X − 4 = − 3 X + 9 ?A. The { Y $}$-coordinates Of The { 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 $}$-coordinates Of

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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 focus on solving the quadratic equation $x^2 - 2x - 4 = -3x + 9$ and find its 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 first simplify it by moving all the terms to one side of the equation.

Simplifying the Equation

We can simplify the equation by adding $3x$ to both sides of the equation, which gives us:

$x^2 - 2x - 4 + 3x = -3x + 9 + 3x$

This simplifies to:

$x^2 + x - 4 = 9$

Rearranging the Equation

Next, we can rearrange the equation by subtracting $9$ from both sides of the equation, which gives us:

$x^2 + x - 13 = 0$

Solving the Quadratic Equation

Now that we have simplified the equation, we can use the quadratic formula to solve for $x$. The quadratic formula is given by:

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

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

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

Calculating the Solutions

Simplifying the expression under the square root, we get:

$x = \frac{-1 \pm \sqrt{1 + 52}}{2}$

$x = \frac{-1 \pm \sqrt{53}}{2}$

The Two Solutions

Therefore, the two solutions of the equation $x^2 - 2x - 4 = -3x + 9$ are:

$x = \frac{-1 + \sqrt{53}}{2}$

$x = \frac{-1 - \sqrt{53}}{2}$

Conclusion

In this article, we have solved the quadratic equation $x^2 - 2x - 4 = -3x + 9$ and found its two solutions. The solutions are given by the quadratic formula, and they are:

$x = \frac{-1 + \sqrt{53}}{2}$

$x = \frac{-1 - \sqrt{53}}{2}$

These solutions can be used to find the $x$-coordinates of the points of intersection of the graphs of $y = x^2 - 2x - 4$ and $y = -3x + 9$.

Discussion

The solutions of the quadratic equation can be used to find the $x$-coordinates of the points of intersection of the graphs of $y = x^2 - 2x - 4$ and $y = -3x + 9$. This can be done by setting the two equations equal to each other and solving for $x$.

The $x$-coordinates of the Points of Intersection

To find the $x$-coordinates of the points of intersection, we can set the two equations equal to each other and solve for $x$. This gives us:

$x^2 - 2x - 4 = -3x + 9$

Simplifying the equation, we get:

$x^2 + x - 13 = 0$

This is the same equation that we solved earlier, and the solutions are:

$x = \frac{-1 + \sqrt{53}}{2}$

$x = \frac{-1 - \sqrt{53}}{2}$

Therefore, the $x$-coordinates of the points of intersection of the graphs of $y = x^2 - 2x - 4$ and $y = -3x + 9$ are:

$x = \frac{-1 + \sqrt{53}}{2}$

$x = \frac{-1 - \sqrt{53}}{2}$

The $y$-coordinates of the Points of Intersection

To find the $y$-coordinates of the points of intersection, we can plug the $x$-coordinates into either of the two equations. Let's use the equation $y = x^2 - 2x - 4$.

Plugging in the $x$-coordinates, we get:

$y = \left(\frac{-1 + \sqrt{53}}{2}\right)^2 - 2\left(\frac{-1 + \sqrt{53}}{2}\right) - 4$

$y = \left(\frac{-1 - \sqrt{53}}{2}\right)^2 - 2\left(\frac{-1 - \sqrt{53}}{2}\right) - 4$

Simplifying the expressions, we get:

$y = \frac{1 - 2\sqrt{53} + 53}{4} + 1 + \sqrt{53} - 4$

$y = \frac{1 + 2\sqrt{53} + 53}{4} + 1 - \sqrt{53} - 4$

Conclusion

In this article, we have solved the quadratic equation $x^2 - 2x - 4 = -3x + 9$ and found its two solutions. We have also found the $x$-coordinates and $y$-coordinates of the points of intersection of the graphs of $y = x^2 - 2x - 4$ and $y = -3x + 9$. The solutions can be used to find the points of intersection of the two graphs.

Final Answer

The final answer is:

$x = \frac{-1 + \sqrt{53}}{2}$

$x = \frac{-1 - \sqrt{53}}{2}$

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Introduction

In our previous article, we solved the quadratic equation $x^2 - 2x - 4 = -3x + 9$ and found its two solutions. In this article, we will answer some frequently asked questions related to solving quadratic equations.

Q1: 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 in the form of $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants.

Q2: How do I solve a quadratic equation?

To solve a quadratic equation, you can use the quadratic formula, which is given by:

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

You can also use factoring, completing the square, or graphing to solve quadratic equations.

Q3: What is the quadratic formula?

The quadratic formula is a formula that gives the solutions to a quadratic equation. It is given by:

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

Q4: How do I use the quadratic formula?

To use the quadratic formula, you need to plug in the values of $a$, $b$, and $c$ into the formula. Then, simplify the expression under the square root and solve for $x$.

Q5: What are the two solutions of the equation $x^2 - 2x - 4 = -3x + 9$?

The two solutions of the equation $x^2 - 2x - 4 = -3x + 9$ are:

$x = \frac{-1 + \sqrt{53}}{2}$

$x = \frac{-1 - \sqrt{53}}{2}$

Q6: How do I find the $x$-coordinates of the points of intersection of the graphs of $y = x^2 - 2x - 4$ and $y = -3x + 9$?

To find the $x$-coordinates of the points of intersection, you can set the two equations equal to each other and solve for $x$. This gives you the solutions to the quadratic equation.

Q7: How do I find the $y$-coordinates of the points of intersection of the graphs of $y = x^2 - 2x - 4$ and $y = -3x + 9$?

To find the $y$-coordinates of the points of intersection, you can plug the $x$-coordinates into either of the two equations.

Q8: What are the $x$-coordinates and $y$-coordinates of the points of intersection of the graphs of $y = x^2 - 2x - 4$ and $y = -3x + 9$?

The $x$-coordinates and $y$-coordinates of the points of intersection are:

$x = \frac{-1 + \sqrt{53}}{2}$

$y = \frac{1 - 2\sqrt{53} + 53}{4} + 1 + \sqrt{53} - 4$

$x = \frac{-1 - \sqrt{53}}{2}$

$y = \frac{1 + 2\sqrt{53} + 53}{4} + 1 - \sqrt{53} - 4$

Conclusion

In this article, we have answered some frequently asked questions related to solving quadratic equations. We have also provided the solutions to the quadratic equation $x^2 - 2x - 4 = -3x + 9$ and the $x$-coordinates and $y$-coordinates of the points of intersection of the graphs of $y = x^2 - 2x - 4$ and $y = -3x + 9$.

Final Answer

The final answer is:

$x = \frac{-1 + \sqrt{53}}{2}$

$x = \frac{-1 - \sqrt{53}}{2}$

$y = \frac{1 - 2\sqrt{53} + 53}{4} + 1 + \sqrt{53} - 4$

$y = \frac{1 + 2\sqrt{53} + 53}{4} + 1 - \sqrt{53} - 4$