The Roots Of A Quadratic Equation Are Given By $x = \frac{-10 \pm \sqrt{100 - 4k^2}}{2k}$. Calculate The Values Of $k$ For Which The Roots Are:1.3.1 Equal1.3.2 Non-real---Simplify The Following Expressions Completely:2.1.1

Introduction

In algebra, 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. The roots of a quadratic equation are the values of $x$ that satisfy the equation. In this article, we will explore the roots of a quadratic equation given by $x = \frac{-10 \pm \sqrt{100 - 4k^2}}{2k}$ and calculate the values of $k$ for which the roots are equal and non-real.

Equal Roots

To find the values of $k$ for which the roots are equal, we need to set the two roots equal to each other and solve for $k$. The two roots are given by $x_1 = \frac{-10 + \sqrt{100 - 4k^2}}{2k}$ and $x_2 = \frac{-10 - \sqrt{100 - 4k^2}}{2k}$.

Setting the two roots equal to each other, we get:

$$\frac{-10 + \sqrt{100 - 4k^2}}{2k} = \frac{-10 - \sqrt{100 - 4k^2}}{2k}$$

Simplifying the equation, we get:

$$\sqrt{100 - 4k^2} = 0$$

Squaring both sides of the equation, we get:

$$100 - 4k^2 = 0$$

Solving for $k$, we get:

$$4k^2 = 100$$

$$k^2 = 25$$

$$k = \pm 5$$

Therefore, the values of $k$ for which the roots are equal are $k = 5$ and $k = -5$.

Non-Real Roots

To find the values of $k$ for which the roots are non-real, we need to ensure that the discriminant $100 - 4k^2$ is negative. The discriminant is given by:

$$100 - 4k^2 < 0$$

Simplifying the inequality, we get:

$$4k^2 > 100$$

Dividing both sides of the inequality by 4, we get:

$$k^2 > 25$$

Taking the square root of both sides of the inequality, we get:

$$k > 5$$

or

$$k < -5$$

Therefore, the values of $k$ for which the roots are non-real are $k > 5$ and $k < -5$.

Simplifying Expressions

2.1.1 Simplify the following expressions completely:

$$\frac{1}{\sqrt{2} + 1}$$

To simplify the expression, we can multiply the numerator and denominator by the conjugate of the denominator:

$$\frac{1}{\sqrt{2} + 1} \cdot \frac{\sqrt{2} - 1}{\sqrt{2} - 1}$$

Simplifying the expression, we get:

$$\frac{\sqrt{2} - 1}{(\sqrt{2} + 1)(\sqrt{2} - 1)}$$

Simplifying the denominator, we get:

$$\frac{\sqrt{2} - 1}{2 - 1}$$

Simplifying the expression, we get:

$$\sqrt{2} - 1$$

Therefore, the simplified expression is $\sqrt{2} - 1$.

Conclusion

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Introduction

In our previous article, we explored the roots of a quadratic equation given by $x = \frac{-10 \pm \sqrt{100 - 4k^2}}{2k}$ and calculated the values of $k$ for which the roots are equal and non-real. In this article, we will answer some frequently asked questions related to the roots of a quadratic equation.

Q&A

Q: What is the general form of a quadratic equation?

A: The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants.

Q: What are the roots of a quadratic equation?

A: The roots of a quadratic equation are the values of $x$ that satisfy the equation.

Q: How do you find the values of $k$ for which the roots are equal?

A: To find the values of $k$ for which the roots are equal, you need to set the two roots equal to each other and solve for $k$.

Q: How do you find the values of $k$ for which the roots are non-real?

A: To find the values of $k$ for which the roots are non-real, you need to ensure that the discriminant $100 - 4k^2$ is negative.

Q: What is the discriminant of a quadratic equation?

A: The discriminant of a quadratic equation is the expression under the square root in the quadratic formula, which is $b^2 - 4ac$.

Q: How do you simplify the expression $\frac{1}{\sqrt{2} + 1}$?

A: To simplify the expression $\frac{1}{\sqrt{2} + 1}$, you can multiply the numerator and denominator by the conjugate of the denominator, which is $\sqrt{2} - 1$.

Q: What is the simplified form of the expression $\frac{1}{\sqrt{2} + 1}$?

A: The simplified form of the expression $\frac{1}{\sqrt{2} + 1}$ is $\sqrt{2} - 1$.

Q: What are the values of $k$ for which the roots are equal?

A: The values of $k$ for which the roots are equal are $k = 5$ and $k = -5$.

Q: What are the values of $k$ for which the roots are non-real?

A: The values of $k$ for which the roots are non-real are $k > 5$ and $k < -5$.

Conclusion

In conclusion, we have answered some frequently asked questions related to the roots of a quadratic equation. We have also provided the general form of a quadratic equation, the definition of the roots of a quadratic equation, and the values of $k$ for which the roots are equal and non-real. We hope that this article has been helpful in clarifying any doubts you may have had about the roots of a quadratic equation.

Additional Resources

• Quadratic Equation Formula
• Discriminant of a Quadratic Equation
• Simplifying Expressions

Final Thoughts

We hope that this article has been helpful in providing a comprehensive overview of the roots of a quadratic equation. If you have any further questions or need additional clarification, please don't hesitate to ask.