The Root Of A Quadratic Equation Is Given By:$ X = -\frac{2 \pm \sqrt{6 - 20ic}}{2} $For Which Value(s) Of $ Ic $ Will The Equation Not Have Real Roots?

Introduction

In mathematics, 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, and $x$ is the variable. The roots of a quadratic equation are the values of $x$ that satisfy the equation. In this article, we will explore the root of a quadratic equation given by $x = -\frac{2 \pm \sqrt{6 - 20ic}}{2}$ and determine for which value(s) of $ic$ the equation will not have real roots.

The Quadratic Formula

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

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

In our case, the quadratic equation is $x = -\frac{2 \pm \sqrt{6 - 20ic}}{2}$. Comparing this with the quadratic formula, we can see that $a = 1$, $b = 0$, and $c = -6 + 20ic$.

The Condition for Real Roots

For a quadratic equation to have real roots, the discriminant, which is the expression under the square root in the quadratic formula, must be non-negative. In other words, the discriminant must satisfy the condition:

$$b^2 - 4ac \geq 0$$

Substituting the values of $a$, $b$, and $c$ into this condition, we get:

$$0^2 - 4(1)(-6 + 20ic) \geq 0$$

Simplifying this expression, we get:

$$24 - 80ic \geq 0$$

The Condition for Complex Roots

For a quadratic equation to have complex roots, the discriminant must be negative. In other words, the discriminant must satisfy the condition:

$$b^2 - 4ac < 0$$

Substituting the values of $a$, $b$, and $c$ into this condition, we get:

$$0^2 - 4(1)(-6 + 20ic) < 0$$

Simplifying this expression, we get:

$$24 - 80ic < 0$$

Solving for ic

To find the value(s) of $ic$ for which the equation will not have real roots, we need to solve the inequality:

$$24 - 80ic < 0$$

Adding $80ic$ to both sides of the inequality, we get:

$$24 < 80ic$$

Dividing both sides of the inequality by $80i$, we get:

$$\frac{24}{80i} < c$$

Simplifying this expression, we get:

$$-\frac{3}{10i} < c$$

Multiplying both sides of the inequality by $-10i$, we get:

$$3i > 10c$$

Dividing both sides of the inequality by $10$, we get:

$$\frac{3i}{10} > c$$

Conclusion

In conclusion, the quadratic equation $x = -\frac{2 \pm \sqrt{6 - 20ic}}{2}$ will not have real roots for values of $ic$ that satisfy the inequality:

$$\frac{3i}{10} > c$$

This means that the equation will have complex roots for values of $ic$ that are greater than $\frac{3i}{10}$.

References

• [1] "Quadratic Equation" by Math Is Fun. Retrieved from https://www.mathisfun.com/algebra/quadratic-equation.html
• [2] "Quadratic Formula" by Khan Academy. Retrieved from https://www.khanacademy.org/math/algebra/quadratic-equations/quadratic-formula/v/quadratic-formula

Further Reading

• [1] "Complex Numbers" by Math Is Fun. Retrieved from https://www.mathisfun.com/algebra/complex-numbers.html
• [2] "Quadratic Equations with Complex Roots" by Khan Academy. Retrieved from https://www.khanacademy.org/math/algebra/quadratic-equations/quadratic-equations-with-complex-roots/v/quadratic-equations-with-complex-roots
  Quadratic Equation Q&A: Understanding the Root of a Quadratic Equation ====================================================================

Introduction

In our previous article, we explored the root of a quadratic equation given by $x = -\frac{2 \pm \sqrt{6 - 20ic}}{2}$ and determined for which value(s) of $ic$ the equation will not have real roots. In this article, we will answer some frequently asked questions related to quadratic equations and provide additional insights into the topic.

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

Q: What is the quadratic formula?

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

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

Q: What is the discriminant?

The discriminant is the expression under the square root in the quadratic formula. It is given by:

$$b^2 - 4ac$$

Q: What is the condition for real roots?

For a quadratic equation to have real roots, the discriminant must be non-negative. In other words, the discriminant must satisfy the condition:

$$b^2 - 4ac \geq 0$$

Q: What is the condition for complex roots?

For a quadratic equation to have complex roots, the discriminant must be negative. In other words, the discriminant must satisfy the condition:

$$b^2 - 4ac < 0$$

Q: How do I determine if a quadratic equation has real or complex roots?

To determine if a quadratic equation has real or complex roots, you need to calculate the discriminant and check if it is non-negative or negative.

Q: What is the significance of the quadratic formula?

The quadratic formula is a powerful tool for solving quadratic equations. It provides a general solution to quadratic equations and can be used to find the roots of any quadratic equation.

Q: Can you provide an example of a quadratic equation with real roots?

Yes, here is an example of a quadratic equation with real roots:

$$x^2 + 4x + 4 = 0$$

To find the roots of this equation, we can use the quadratic formula:

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

Substituting the values of $a$, $b$, and $c$ into this formula, we get:

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

Simplifying this expression, we get:

$$x = \frac{-4 \pm \sqrt{0}}{2}$$

This means that the equation has two real roots, $x = -2$.

Q: Can you provide an example of a quadratic equation with complex roots?

Yes, here is an example of a quadratic equation with complex roots:

$$x^2 + 4x + 5 = 0$$

To find the roots of this equation, we can use the quadratic formula:

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

Substituting the values of $a$, $b$, and $c$ into this formula, we get:

$$x = \frac{-4 \pm \sqrt{4^2 - 4(1)(5)}}{2(1)}$$

Simplifying this expression, we get:

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

This means that the equation has two complex roots, $x = -2 \pm i$.

Conclusion

In conclusion, quadratic equations are a fundamental concept in mathematics, and understanding the root of a quadratic equation is crucial for solving quadratic equations. We hope that this Q&A article has provided you with a better understanding of quadratic equations and their roots. If you have any further questions or need additional clarification, please don't hesitate to ask.