Understanding Real and Distinct Roots

Oct 13, 2025 by windailys.com 35 views

Introduction

In algebra, the roots of a quadratic equation are the values of the variable that satisfy the equation. A quadratic equation is in the form of ${ax^2 + bx + c = 0}$ , where ${a}$ , ${b}$ , and ${c}$ are constants. The roots of a quadratic equation can be real or complex, and they can be distinct or repeated. In this article, we will focus on finding the values of ${p}$ for which the equation ${2x^2 - (1+p)x + 5 = p}$ has real and distinct roots.

Understanding Real and Distinct Roots

Real and distinct roots are the roots of a quadratic equation that are real numbers and are not equal to each other. In other words, the roots are distinct and can be written as ${x_1}$ and ${x_2}$ , where ${x_1 \neq x_2}$ . For a quadratic equation to have real and distinct roots, the discriminant, which is the expression under the square root in the quadratic formula, must be greater than zero.

The Quadratic Formula

The quadratic formula is a formula that gives the roots of a quadratic equation. It is given by:

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

In this formula, ${a}$ , ${b}$ , and ${c}$ are the constants in the quadratic equation, and ${x}$ is the root of the equation. The discriminant is the expression under the square root, which is ${b^2 - 4ac}$ .

Applying the Quadratic Formula to the Given Equation

To find the values of ${p}$ for which the equation ${2x^2 - (1+p)x + 5 = p}$ has real and distinct roots, we need to apply the quadratic formula to the equation. First, we need to rewrite the equation in the standard form of a quadratic equation, which is ${ax^2 + bx + c = 0}$ . We can do this by subtracting ${p}$ from both sides of the equation, which gives us:

${2x^2 - (1+p)x + (5-p) = 0}$

Now, we can identify the values of ${a}$ , ${b}$ , and ${c}$ in the equation. We have ${a = 2}$ , ${b = -(1+p)}$ , and ${c = 5-p}$ .

Finding the Discriminant

To find the values of ${p}$ for which the equation has real and distinct roots, we need to find the discriminant, which is the expression under the square root in the quadratic formula. The discriminant is given by:

${b^2 - 4ac}$

Substituting the values of ${a}$ , ${b}$ , and ${c}$ into the expression, we get:

${(-(1+p))^2 - 4(2)(5-p)}$

Expanding and simplifying the expression, we get:

${(1+p)^2 - 40 + 8p}$

Expanding the square, we get:

${1 + 2p + p^2 - 40 + 8p}$

Combining like terms, we get:

${p^2 + 10p - 39}$

Solving the Inequality

To find the values of ${p}$ for which the equation has real and distinct roots, we need to solve the inequality:

${p^2 + 10p - 39 > 0}$

This is a quadratic inequality, and we can solve it by factoring the left-hand side of the inequality. We can factor the quadratic expression as:

${(p + 13)(p - 3) > 0}$

Finding the Intervals of p

To solve the inequality, we need to find the intervals of ${p}$ for which the inequality is true. We can do this by finding the values of ${p}$ for which the expression ${(p + 13)(p - 3)}$ is positive.

The Solution

The solution to the inequality is:

${p < -13 \text{ or } p > 3}$

This means that the values of ${p}$ for which the equation ${2x^2 - (1+p)x + 5 = p}$ has real and distinct roots are ${p < -13}$ or ${p > 3}$ .

Conclusion

In this article, we have found the values of ${p}$ for which the equation ${2x^2 - (1+p)x + 5 = p}$ has real and distinct roots. We have applied the quadratic formula to the equation, found the discriminant, and solved the inequality to find the values of ${p}$ . The solution to the inequality is ${p < -13}$ or ${p > 3}$ .

Q: What is the significance of real and distinct roots in a quadratic equation?

A: Real and distinct roots are the roots of a quadratic equation that are real numbers and are not equal to each other. In other words, the roots are distinct and can be written as ${x_1}$ and ${x_2}$ , where ${x_1 \neq x_2}$ . For a quadratic equation to have real and distinct roots, the discriminant, which is the expression under the square root in the quadratic formula, must be greater than zero.

Q: What is the discriminant in a quadratic equation?

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

${b^2 - 4ac}$

Q: How do you find the discriminant in a quadratic equation?

A: To find the discriminant, you need to identify the values of ${a}$ , ${b}$ , and ${c}$ in the quadratic equation. Then, you can substitute these values into the expression for the discriminant.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that gives the roots of a quadratic equation. It is given by:

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

Q: How do you apply the quadratic formula to a quadratic equation?

A: To apply the quadratic formula, you need to identify the values of ${a}$ , ${b}$ , and ${c}$ in the quadratic equation. Then, you can substitute these values into the quadratic formula.

Q: What is the significance of the inequality ${p^2 + 10p - 39 > 0}$ in finding the values of p for real and distinct roots?

A: The inequality ${p^2 + 10p - 39 > 0}$ is a quadratic inequality that represents the values of ${p}$ for which the equation ${2x^2 - (1+p)x + 5 = p}$ has real and distinct roots.

Q: How do you solve the inequality ${p^2 + 10p - 39 > 0}$ ?

A: To solve the inequality, you need to factor the left-hand side of the inequality. You can factor the quadratic expression as:

${(p + 13)(p - 3) > 0}$

Q: What is the solution to the inequality ${p^2 + 10p - 39 > 0}$ ?

A: The solution to the inequality is:

${p < -13 \text{ or } p > 3}$

This means that the values of ${p}$ for which the equation ${2x^2 - (1+p)x + 5 = p}$ has real and distinct roots are ${p < -13}$ or ${p > 3}$ .

Q: What are the implications of the solution to the inequality ${p^2 + 10p - 39 > 0}$ ?

A: The solution to the inequality implies that the values of ${p}$ for which the equation ${2x^2 - (1+p)x + 5 = p}$ has real and distinct roots are ${p < -13}$ or ${p > 3}$ . This means that the equation has real and distinct roots for any value of ${p}$ that satisfies this condition.

Q: How do you apply the solution to the inequality ${p^2 + 10p - 39 > 0}$ in a real-world context?

A: The solution to the inequality can be applied in a real-world context by considering the values of ${p}$ that satisfy the condition. For example, if you are working with a quadratic equation that represents a physical system, you can use the solution to the inequality to determine the values of ${p}$ that correspond to real and distinct roots.

Q: What are some common applications of the quadratic formula and the inequality ${p^2 + 10p - 39 > 0}$ ?

A: The quadratic formula and the inequality ${p^2 + 10p - 39 > 0}$ have many applications in mathematics, physics, engineering, and other fields. Some common applications include:

Finding the roots of a quadratic equation
Determining the stability of a physical system
Modeling population growth and decay
Analyzing the behavior of electrical circuits
Solving optimization problems

Q: How do you extend the solution to the inequality ${p^2 + 10p - 39 > 0}$ to more complex quadratic equations?

A: To extend the solution to the inequality ${p^2 + 10p - 39 > 0}$ to more complex quadratic equations, you can use the quadratic formula and the inequality in a similar way. You can also use other techniques, such as graphing and numerical methods, to find the roots of the quadratic equation.

Q: What are some common mistakes to avoid when applying the quadratic formula and the inequality ${p^2 + 10p - 39 > 0}$ ?

A: Some common mistakes to avoid when applying the quadratic formula and the inequality ${p^2 + 10p - 39 > 0}$ include:

Failing to identify the values of ${a}$ , ${b}$ , and ${c}$ in the quadratic equation
Making errors in the calculation of the discriminant
Failing to consider the sign of the discriminant
Not checking the validity of the solution to the inequality

Q: How do you verify the solution to the inequality ${p^2 + 10p - 39 > 0}$ ?

A: To verify the solution to the inequality ${p^2 + 10p - 39 > 0}$ , you can use a variety of methods, including:

Graphing the quadratic function
Using numerical methods to find the roots of the quadratic equation
Checking the sign of the discriminant
Verifying the solution using a different method, such as the quadratic formula.