Problem 68 - College Algebra

Question

Solve each equation in Exercises \(65-74\) using the quadratic formula. $$x^{2}+5 x+2=0$$

Step-by-Step Solution

Answer

The solutions for the equation \(x^{2}+5x+2=0\) are \(x1 = \frac{-5 + \sqrt{17}}{2}\) and \(x2 = \frac{-5 - \sqrt{17}}{2}\)

Step 1: Identify the Coefficients a, b, and c

In the quadratic equation \(x^{2}+5x+2=0\), the coefficients are \(a = 1\), \(b = 5\), and \(c = 2\).

Step 2: Calculate the Discriminant

The discriminant is given by \(b^{2}- 4ac\). Substituting the earlier identified coefficients, we get \(5^{2} - 4*1*2 = 17\).

Step 3: Apply the Quadratic Formula

To find the roots of the equation, use the quadratic formula \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\). Substituting our coefficients and the discriminant, we get \(x = \frac{-5\pm \sqrt{17}}{2}\). There are thus two solutions for x: \(x1 = \frac{-5 + \sqrt{17}}{2}\) and \(x2 = \frac{-5 - \sqrt{17}}{2}\) .

Key Concepts

Solving Quadratic Equations

Quadratic equations are equations of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(x\) represents the unknown variable.
To solve these equations, one of the most effective methods is using the quadratic formula. The quadratic formula is:

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

This formula provides a straightforward way to find the roots, or solutions, of the quadratic equation.
The solutions \(x_1\) and \(x_2\) are found by substituting the coefficients \(a\), \(b\), and \(c\) into the formula.
By understanding the quadratic formula, you can easily determine the values of \(x\) that satisfy the equation.

Discriminant

The discriminant is a key part of the quadratic formula, found within the square root: \(b^2 - 4ac\).
The discriminant helps to determine the nature of the roots of a quadratic equation. It is a major component in understanding how many and what type of solutions the equation has.

• If \(b^2 - 4ac > 0\): The equation has two distinct real roots.
• If \(b^2 - 4ac = 0\): There is exactly one real root, or a repeated root.
• If \(b^2 - 4ac < 0\): The equation has two complex roots.

In the equation \(x^2 + 5x + 2 = 0\), the discriminant is \(17\), indicating two distinct real roots.
Recognizing the value and significance of the discriminant can streamline the process of solving quadratic equations and predicting the outcomes.

Quadratic Roots

Quadratic roots are the solutions for \(x\) in the quadratic equation \(ax^2 + bx + c = 0\).
These roots can be real or complex, depending on the discriminant. Finding the roots is essential because they represent the values of \(x\) that make the equation true.
Applying the quadratic formula provides direct access to the roots:

• With the example equation, \(x^2 + 5x + 2 = 0\), using our formula, the roots are: \( x_1 = \frac{-5 + \sqrt{17}}{2} \) and \( x_2 = \frac{-5 - \sqrt{17}}{2} \).

Hence, we find that \(x_1\) and \(x_2\) are the solutions that satisfy the equation.
Understanding how to compute these roots is crucial, as it allows for solving a variety of problems involving quadratic equations.

Coefficients in Quadratic Equations

Coefficients in a quadratic equation are the numbers \(a\), \(b\), and \(c\) in the equation \(ax^2 + bx + c = 0\).
These values are critical as they determine the shape and position of the parabola represented by the quadratic equation.
To effectively utilize the quadratic formula, it is vital to correctly identify these coefficients:

• \(a\) is the coefficient of \(x^2\) and affects the parabolic opening's width and direction.
• \(b\) is the coefficient of \(x\), influencing the parabola's symmetry.
• \(c\) is the constant term, determining the y-intercept of the graph.

In the quadratic equation \(x^2 + 5x + 2 = 0\), the coefficients are \(a = 1\), \(b = 5\), and \(c = 2\).
Correctly identifying these coefficients ensures accurate calculation of the roots using the quadratic formula.