Question

Find the real solutions, if any, of each equation. Use the quadratic formula. $$ x^{2}+4 x+2=0 $$

Short answer

\[ x = -2 \pm \sqrt{2} \]

Step-by-step solution

Identify Coefficients

The equation is in the form of a quadratic equation, which is given by ax^{2} + bx + c = 0a = 1, b = 4, and c = 2

Write the Quadratic Formula

The quadratic formula is used to find the solutions to the quadratic equation. The formula is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Calculate the Discriminant

The discriminant Δ is given by:b^2 - 4acSubstitute the values of a, b, and c:Δ = 4^2 - 4(1)(2)Δ = 16 - 8Δ = 8

Apply the Quadratic Formula

Substitute the values of a, b, and Δ into the quadratic formula:\[ x = \frac{-4 \pm \sqrt{8}}{2(1)} \] \[ x = \frac{-4 \pm 2\sqrt{2}}{2} \] \[ x = -2 \pm \sqrt{2} \]

Write the Solutions

The solutions to the quadratic equation are:\[ x = -2 + \sqrt{2} \] \[ x = -2 - \sqrt{2} \]

Key concepts

quadratic equations

Quadratic equations are a type of polynomial equation where the highest exponent of the variable is 2. They typically take the form:

• \( ax^{2} + bx + c = 0 \)

where a, b, and c are coefficients and x represents the variable.

Quadratic equations are fundamental in algebra and often arise in various math and science applications.
These equations can have up to two solutions, depending on the values of a, b, and c.
To solve a quadratic equation, you can use methods such as:

• Factoring
• Completing the square
• Graphing
• Using the quadratic formula

The quadratic formula is a powerful tool that works for all quadratic equations, making it a reliable method for finding solutions.

discriminant

The discriminant is a key component in determining the nature of the solutions for a quadratic equation.
The discriminant, denoted by Δ, is calculated using the formula:

• \( \Delta = b^2 - 4ac \)

It is derived from the quadratic formula and appears under the square root.

The value of the discriminant reveals important information about the solutions of the quadratic equation:

• If \( \Delta > 0 \) , there are two distinct real solutions
• If \( \Delta = 0 \) , there is exactly one real solution (a repeated root)
• If \( \Delta < 0 \) , there are no real solutions (only complex or imaginary solutions)

In the given exercise, the discriminant is computed as follows:

• \( 4^2 - 4(1)(2) = 16 - 8 = 8 \)

Since \( 8 \) is greater than 0, we know there are two distinct real solutions.

real solutions

Real solutions are the values of x that satisfy the quadratic equation and are real numbers.
These solutions can be found using the quadratic formula:

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

To determine if the solutions are real, we first compute the discriminant:

• \( \Delta = b^2 - 4ac \)

For the exercise \( x^2 + 4x + 2 = 0 \) , we calculated the discriminant and found it to be positive, which means there are two real solutions.
We substitute the coefficients and the discriminant back into the quadratic formula to find the solutions:

• \[ x = \frac{-4 \pm \sqrt{8}}{2(1)} \] \[ x = \frac{-4 \pm 2\sqrt{2}}{2} \] \[ x = -2 \pm \sqrt{2} \]

Thus, the real solutions to the equation are:

• \( x = -2 + \sqrt{2} \)
• \(x = -2 - \sqrt{2} \)

These solutions can be verified by substituting them back into the original equation.