Verified Answer :

  • Identify the coefficients of the quadratic equation \(x^2 + 6x + 8 = 0\) as \(a = 1, b = 6, c = 8\).
  • Calculate the discriminant \(D = b^2 - 4ac = 4\).
  • Apply the quadratic formula \(x = \frac{-b \pm \sqrt{D}}{2a}\) to find the roots.
  • The solution set for the equation is \(\boxed{\{-4, -2\}}\).

Explanation

  1. Analyzing the Equation and the Quadratic Formula

The quadratic equation provided is \(x^2 + 6x + 8 = 0\). This is a standard quadratic equation of the form \(ax^2 + bx + c = 0\), where the coefficients are \(a = 1\), \(b = 6\), and \(c = 8\). Our goal is to solve for \(x\) using the quadratic formula, which is given by:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
This formula provides the values of \(x\) that satisfy the equation.

  1. Calculating the Discriminant

To apply the quadratic formula, we first calculate the discriminant, which is the expression inside the square root, \(D = b^2 - 4ac\). Substituting our coefficients \(a = 1\), \(b = 6\), and \(c = 8\):
\[D = (6)^2 - 4(1)(8)\]
\[D = 36 - 32\]
\[D = 4\]
Since the discriminant is positive, the equation has two distinct real roots.

  1. Finding the Roots

Now, we substitute the discriminant back into the quadratic formula:
\[x = \frac{-6 \pm \sqrt{4}}{2(1)}\]
Since \(\sqrt{4} = 2\), the expression simplifies to:
\[x = \frac{-6 \pm 2}{2}\]
We can now calculate the two possible values for \(x\):

  1. The first solution is \(x_1 = \frac{-6 + 2}{2} = \frac{-4}{2} = -2\).
  2. The second solution is \(x_2 = \frac{-6 - 2}{2} = \frac{-8}{2} = -4\).

Thus, the solutions to the equation are \(x = -2\) and \(x = -4\).

  1. Final Answer

The roots of the equation \(x^2 + 6x + 8 = 0\) are \(-2\) and \(-4\). Therefore, the solution set is \(\{-4, -2\}\).
\[\boxed{\{-4, -2\}}\]

Examples

Quadratic equations are widely used in physics and engineering to model projectile motion. For example, if you throw a ball, its height over time is described by a quadratic equation. By solving the equation for when the height is zero, you can determine exactly when the ball will hit the ground. Understanding how to use the quadratic formula allows you to find these critical time values easily, which is essential for predicting the trajectory and landing point of any launched object.

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