Solve The Inequality:$\[ Y \ \textless \ -2x^2 + 8x - 4 \\]

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Introduction

In this article, we will delve into the world of inequalities and learn how to solve a quadratic inequality of the form $y < -2x^2 + 8x - 4$. Inequalities are a fundamental concept in mathematics, and solving them requires a deep understanding of algebraic manipulations and graphing techniques. By the end of this article, you will be able to solve quadratic inequalities of this form and understand the graphical representation of the solution set.

Understanding the Inequality

The given inequality is $y < -2x^2 + 8x - 4$. To solve this inequality, we need to find the values of $x$ for which the expression $-2x^2 + 8x - 4$ is less than $y$. This means that we need to find the values of $x$ that make the expression $-2x^2 + 8x - 4$ negative.

Factoring the Quadratic Expression

To solve the inequality, we first need to factor the quadratic expression $-2x^2 + 8x - 4$. We can start by factoring out the common factor of $-2$ from the expression:

$$-2x^2 + 8x - 4 = -2(x^2 - 4x + 2)$$

Next, we can try to factor the quadratic expression inside the parentheses:

$$x^2 - 4x + 2 = (x - 2)^2 - 2$$

However, this expression cannot be factored further. Therefore, we will use the quadratic formula to find the roots of the expression.

Using the Quadratic Formula

The quadratic formula is given by:

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

In this case, $a = -2$, $b = 8$, and $c = -4$. Plugging these values into the formula, we get:

$$x = \frac{-8 \pm \sqrt{8^2 - 4(-2)(-4)}}{2(-2)}$$

Simplifying the expression, we get:

$$x = \frac{-8 \pm \sqrt{64 - 32}}{-4}$$

$$x = \frac{-8 \pm \sqrt{32}}{-4}$$

$$x = \frac{-8 \pm 4\sqrt{2}}{-4}$$

$$x = 2 \pm \sqrt{2}$$

Graphing the Solution Set

To graph the solution set, we need to find the values of $x$ for which the expression $-2x^2 + 8x - 4$ is less than $y$. We can do this by plotting the graph of the expression $-2x^2 + 8x - 4$ and finding the values of $x$ for which the graph is below the $x$-axis.

The graph of the expression $-2x^2 + 8x - 4$ is a parabola that opens downward. The vertex of the parabola is at the point $(2, -4)$. The parabola intersects the $x$-axis at the points $(2 + \sqrt{2}, 0)$ and $(2 - \sqrt{2}, 0)$.

Conclusion

In this article, we learned how to solve the inequality $y < -2x^2 + 8x - 4$. We factored the quadratic expression, used the quadratic formula to find the roots, and graphed the solution set. By following these steps, you can solve quadratic inequalities of this form and understand the graphical representation of the solution set.

Final Answer

The final answer is $\boxed{(-\infty, 2 - \sqrt{2}) \cup (2 + \sqrt{2}, \infty)}$.

Step-by-Step Solution

Here is the step-by-step solution to the inequality:

1. Factor the quadratic expression $-2x^2 + 8x - 4$.
2. Use the quadratic formula to find the roots of the expression.
3. Graph the solution set by plotting the graph of the expression $-2x^2 + 8x - 4$ and finding the values of $x$ for which the graph is below the $x$-axis.

Frequently Asked Questions

• What is the solution to the inequality $y < -2x^2 + 8x - 4$?
• How do I factor the quadratic expression $-2x^2 + 8x - 4$?
• What is the quadratic formula, and how do I use it to find the roots of the expression?
• How do I graph the solution set?

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Additional Resources

• Khan Academy: Quadratic Equations
• MIT OpenCourseWare: Calculus
• Wolfram Alpha: Quadratic Formula
  

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Introduction

In our previous article, we learned how to solve the inequality $y < -2x^2 + 8x - 4$. We factored the quadratic expression, used the quadratic formula to find the roots, and graphed the solution set. In this article, we will answer some frequently asked questions about solving the inequality.

Q&A

Q: What is the solution to the inequality $y < -2x^2 + 8x - 4$?

A: The solution to the inequality $y < -2x^2 + 8x - 4$ is $(-\infty, 2 - \sqrt{2}) \cup (2 + \sqrt{2}, \infty)$.

Q: How do I factor the quadratic expression $-2x^2 + 8x - 4$?

A: To factor the quadratic expression $-2x^2 + 8x - 4$, we can start by factoring out the common factor of $-2$ from the expression. We can then try to factor the quadratic expression inside the parentheses.

Q: What is the quadratic formula, and how do I use it to find the roots of the expression?

A: The quadratic formula is given by:

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

To use the quadratic formula, we need to plug in the values of $a$, $b$, and $c$ into the formula. In this case, $a = -2$, $b = 8$, and $c = -4$. Plugging these values into the formula, we get:

$$x = \frac{-8 \pm \sqrt{8^2 - 4(-2)(-4)}}{2(-2)}$$

Simplifying the expression, we get:

$$x = \frac{-8 \pm \sqrt{64 - 32}}{-4}$$

$$x = \frac{-8 \pm \sqrt{32}}{-4}$$

$$x = \frac{-8 \pm 4\sqrt{2}}{-4}$$

$$x = 2 \pm \sqrt{2}$$

Q: How do I graph the solution set?

A: To graph the solution set, we need to find the values of $x$ for which the expression $-2x^2 + 8x - 4$ is less than $y$. We can do this by plotting the graph of the expression $-2x^2 + 8x - 4$ and finding the values of $x$ for which the graph is below the $x$-axis.

Q: What is the vertex of the parabola?

A: The vertex of the parabola is at the point $(2, -4)$.

Q: Where does the parabola intersect the $x$-axis?

A: The parabola intersects the $x$-axis at the points $(2 + \sqrt{2}, 0)$ and $(2 - \sqrt{2}, 0)$.

Conclusion

In this article, we answered some frequently asked questions about solving the inequality $y < -2x^2 + 8x - 4$. We provided step-by-step solutions to the inequality and explained how to factor the quadratic expression, use the quadratic formula, and graph the solution set.

Final Answer

The final answer is $\boxed{(-\infty, 2 - \sqrt{2}) \cup (2 + \sqrt{2}, \infty)}$.

Step-by-Step Solution

Here is the step-by-step solution to the inequality:

1. Factor the quadratic expression $-2x^2 + 8x - 4$.
2. Use the quadratic formula to find the roots of the expression.
3. Graph the solution set by plotting the graph of the expression $-2x^2 + 8x - 4$ and finding the values of $x$ for which the graph is below the $x$-axis.

Frequently Asked Questions

• What is the solution to the inequality $y < -2x^2 + 8x - 4$?
• How do I factor the quadratic expression $-2x^2 + 8x - 4$?
• What is the quadratic formula, and how do I use it to find the roots of the expression?
• How do I graph the solution set?

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Additional Resources

• Khan Academy: Quadratic Equations
• MIT OpenCourseWare: Calculus
• Wolfram Alpha: Quadratic Formula