Equal Roots of a Quadratic Equation

Oct 13, 2025 by miodeya.com 46 views

Introduction

Quadratic equations are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and economics. In this article, we will delve into the world of quadratic equations and explore some of the most common problems and solutions. We will start by discussing the conditions under which the roots of a quadratic equation are equal, and then move on to determining the values of a parameter for which the equation has two unequal real roots.

Equal Roots of a Quadratic Equation

The roots of a quadratic equation $ax^2 + bx + c = 0$ are given by the quadratic formula:

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

For the roots to be equal, the discriminant $b^2 - 4ac$ must be equal to zero. Let's consider the quadratic equation $x^2 - 6x + r = 0$ . In this case, $a = 1$ , $b = -6$ , and $c = r$ . The discriminant is given by:

b^2 - 4ac = (-6)^2 - 4(1)(r) = 36 - 4r

For the roots to be equal, we must have:

36 - 4r = 0

Solving for $r$ , we get:

r = \frac{36}{4} = 9

Therefore, the value of $r$ for which the roots of the quadratic equation $x^2 - 6x + r = 0$ are equal is $r = 9$ .

Unequal Real Roots of a Quadratic Equation

For a quadratic equation to have two unequal real roots, the discriminant must be greater than zero. Let's consider the quadratic equation $-2x^2 + 4x - 3 = -p$ . We can rewrite this equation as:

2x^2 - 4x + (3 + p) = 0

In this case, $a = 2$ , $b = -4$ , and $c = 3 + p$ . The discriminant is given by:

b^2 - 4ac = (-4)^2 - 4(2)(3 + p) = 16 - 8(3 + p)

For the equation to have two unequal real roots, we must have:

16 - 8(3 + p) > 0

Simplifying the inequality, we get:

16 - 24 - 8p > 0

-8 - 8p > 0

-8p > -8

p < \frac{-8}{-8}

p < 1

Therefore, the values of $p$ for which the equation $-2x^2 + 4x - 3 = -p$ will have two unequal real roots are $p < 1$ .

A Quadratic Equation with a Parameter

Consider the quadratic equation $kx^2 + bx + c = 0$ . We can rewrite this equation as:

kx^2 + bx + c = 0

In this case, $a = k$ , $b = b$ , and $c = c$ . The discriminant is given by:

b^2 - 4ac = b^2 - 4kc

For the equation to have two unequal real roots, we must have:

b^2 - 4kc > 0

Simplifying the inequality, we get:

b^2 > 4kc

\frac{b^2}{4k} > c

Therefore, the values of $k$ for which the equation $kx^2 + bx + c = 0$ will have two unequal real roots are $\frac{b^2}{4c} < k$ .

Conclusion

In this article, we have discussed some of the most common problems and solutions related to quadratic equations. We have shown that the roots of a quadratic equation are equal when the discriminant is equal to zero, and we have determined the values of a parameter for which the equation has two unequal real roots. We have also considered a quadratic equation with a parameter and found the values of the parameter for which the equation has two unequal real roots. We hope that this article has provided a comprehensive guide to solving quadratic equations and has helped readers to better understand this important concept in mathematics.

References

[1] "Quadratic Equations" by Math Open Reference
[2] "Quadratic Formula" by Wolfram MathWorld
[3] "Discriminant" by Khan Academy

Glossary

Discriminant: The expression $b^2 - 4ac$ under the square root in the quadratic formula.
Quadratic Formula: The formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ for finding the roots of a quadratic equation.
Quadratic Equation: An equation of the form $ax^2 + bx + c = 0$ .

FAQs

Q: What is the condition for the roots of a quadratic equation to be equal? A: The roots of a quadratic equation are equal when the discriminant is equal to zero.
Q: What is the condition for a quadratic equation to have two unequal real roots? A: A quadratic equation has two unequal real roots when the discriminant is greater than zero.
Q: How do I find the values of a parameter for which a quadratic equation has two unequal real roots? A: To find the values of a parameter for which a quadratic equation has two unequal real roots, you need to find the values of the parameter that make the discriminant greater than zero.

Introduction

Q: What is a Quadratic Equation?

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

Q: What is the Quadratic Formula?

A: The quadratic formula is a formula for finding the roots of a quadratic equation. It is given by:

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

Q: What is the Discriminant?

A: The discriminant is the expression $b^2 - 4ac$ under the square root in the quadratic formula. It is used to determine the nature of the roots of a quadratic equation.

Q: What is the Condition for the Roots of a Quadratic Equation to be Equal?

A: The roots of a quadratic equation are equal when the discriminant is equal to zero, i.e., $b^2 - 4ac = 0$ .

Q: What is the Condition for a Quadratic Equation to have Two Unequal Real Roots?

A: A quadratic equation has two unequal real roots when the discriminant is greater than zero, i.e., $b^2 - 4ac > 0$ .

Q: How Do I Find the Values of a Parameter for Which a Quadratic Equation has Two Unequal Real Roots?

A: To find the values of a parameter for which a quadratic equation has two unequal real roots, you need to find the values of the parameter that make the discriminant greater than zero.

Q: What is the Relationship Between the Coefficients of a Quadratic Equation and Its Roots?

A: The coefficients of a quadratic equation are related to its roots by the following formulas:

x_1 + x_2 = -\frac{b}{a}

x_1x_2 = \frac{c}{a}

Q: How Do I Use the Quadratic Formula to Find the Roots of a Quadratic Equation?

A: To use the quadratic formula to find the roots of a quadratic equation, you need to plug in the values of $a$ , $b$ , and $c$ into the formula and simplify.

Q: What is the Difference Between a Quadratic Equation and a Linear Equation?

A: A quadratic equation is an equation of the form $ax^2 + bx + c = 0$ , while a linear equation is an equation of the form $ax + b = 0$ .

Q: Can a Quadratic Equation Have Three or More Roots?

A: No, a quadratic equation can have at most two roots.

Q: Can a Quadratic Equation Have Complex Roots?

A: Yes, a quadratic equation can have complex roots.

Q: How Do I Determine the Nature of the Roots of a Quadratic Equation?

A: To determine the nature of the roots of a quadratic equation, you need to examine the discriminant. If the discriminant is positive, the equation has two unequal real roots. If the discriminant is zero, the equation has two equal real roots. If the discriminant is negative, the equation has two complex roots.

Q: What is the Significance of the Quadratic Formula in Real-World Applications?

A: The quadratic formula has numerous real-world applications, including physics, engineering, economics, and computer science. It is used to model and solve problems involving quadratic equations.

Q: Can I Use the Quadratic Formula to Solve Non-Quadratic Equations?

A: No, the quadratic formula is only applicable to quadratic equations. It cannot be used to solve non-quadratic equations.

Q: How Do I Simplify a Quadratic Equation?

A: To simplify a quadratic equation, you need to combine like terms and factor out any common factors.

Q: Can I Use the Quadratic Formula to Find the Roots of a Quadratic Equation with Complex Coefficients?

A: No, the quadratic formula is only applicable to quadratic equations with real coefficients. It cannot be used to find the roots of quadratic equations with complex coefficients.

Conclusion

In this article, we have provided a comprehensive Q&A guide to quadratic equations, covering a wide range of topics and questions. We hope that this guide has been helpful in understanding the concept of quadratic equations and their applications in real-world problems.

References

[1] "Quadratic Equations" by Math Open Reference
[2] "Quadratic Formula" by Wolfram MathWorld
[3] "Discriminant" by Khan Academy

Glossary

Discriminant: The expression $b^2 - 4ac$ under the square root in the quadratic formula.
Quadratic Formula: The formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ for finding the roots of a quadratic equation.
Quadratic Equation: An equation of the form $ax^2 + bx + c = 0$ .

FAQs

Q: What is the condition for the roots of a quadratic equation to be equal? A: The roots of a quadratic equation are equal when the discriminant is equal to zero.
Q: What is the condition for a quadratic equation to have two unequal real roots? A: A quadratic equation has two unequal real roots when the discriminant is greater than zero.
Q: How do I find the values of a parameter for which a quadratic equation has two unequal real roots? A: To find the values of a parameter for which a quadratic equation has two unequal real roots, you need to find the values of the parameter that make the discriminant greater than zero.