Find the value of given expression with the help of given quadratic equation.

To find the value of a given expression using a quadratic equation, we first need to understand the relationship between the expression and the roots of the quadratic equation. Let's break this down step by step.

Understanding Quadratic Equations

A quadratic equation is typically expressed in the form:

ax² + bx + c = 0

Here, a, b, and c are constants, and a cannot be zero. The solutions to this equation, known as the roots, can be found using the quadratic formula:

x = (-b ± √(b² - 4ac)) / (2a)

Example Quadratic Equation

Let's consider the quadratic equation:

2x² - 4x - 6 = 0

In this case, a = 2, b = -4, and c = -6.

Finding the Roots

We can apply the quadratic formula to find the roots:

• Calculate the discriminant: b² - 4ac = (-4)² - 4(2)(-6) = 16 + 48 = 64
• Now, substitute into the quadratic formula:
• x = (4 ± √64) / (2 * 2)
• x = (4 ± 8) / 4

This gives us two roots:

• x₁ = (4 + 8) / 4 = 3
• x₂ = (4 - 8) / 4 = -1

Using the Roots to Evaluate the Expression

Now that we have the roots, we can evaluate a specific expression that might depend on these roots. For example, if we want to find the value of the expression:

f(x) = x² + 2x + 1

We can substitute the roots into this expression:

• For x₁ = 3:
  f(3) = 3² + 2(3) + 1 = 9 + 6 + 1 = 16
• For x₂ = -1:
  f(-1) = (-1)² + 2(-1) + 1 = 1 - 2 + 1 = 0

Final Thoughts

In summary, to find the value of a given expression using a quadratic equation, you first determine the roots of the equation using the quadratic formula. Then, you substitute these roots into the expression to find the desired values. This method not only helps in solving the problem but also reinforces the connection between quadratic equations and their applications in various expressions.