Problem 28 - Elementary Algebra

Question

For the following problems, solve the equations using the quadratic formula. $$ x^{2}-3 x+1=0 $$

Step-by-Step Solution

Answer

Answer: The roots of the quadratic equation are \(x_1=\frac{3+\sqrt{5}}{2}\) and \(x_2=\frac{3-\sqrt{5}}{2}\).

Step 1: Identify the coefficients

Given the quadratic equation: \(x^2-3x+1=0\). The coefficients are: \(a=1\), \(b=-3\), and \(c=1\).

Step 2: Write down the quadratic formula

The quadratic formula is given as: $$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$

Step 3: Substitute the coefficients into the formula

Substitute the coefficients \(a=1\), \(b=-3\), and \(c=1\) into the quadratic formula: $$x=\frac{-(-3)\pm\sqrt{(-3)^2-4(1)(1)}}{2(1)}$$

Step 4: Simplify the formula

Simplify the formula: $$x=\frac{3\pm\sqrt{9-4}}{2}$$

Step 5: Calculate the roots

Calculate the roots by solving for both the positive and negative cases: $$\begin{array}{ r l } x_1=&\frac{3+\sqrt{5}}{2}\\ x_2=&\frac{3-\sqrt{5}}{2} \end{array}$$ The roots of the quadratic equation \(x^2-3x+1=0\) are: $$x_1=\frac{3+\sqrt{5}}{2}$$ and $$x_2=\frac{3-\sqrt{5}}{2}$$

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool used to solve quadratic equations, which are equations of the form \(ax^2 + bx + c = 0\). This formula offers a consistent way to find the solutions (or roots) of any quadratic equation without having to factorize or complete the square. The quadratic formula is:

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

In this formula, \(a\), \(b\), and \(c\) are coefficients from the quadratic equation, where \(aeq0\) is the coefficient of \(x^2\), \(b\) is the coefficient of \(x\), and \(c\) is the constant term. The symbol \(\pm\) indicates that the equation has two possible solutions. Understanding the quadratic formula is key for solving many algebra problems effectively. The formula comprises two main parts:

• The numerator part displays \(-b\) and the discriminant \(\sqrt{b^2-4ac}\).
• The denominator part, \(2a\), dictates how the entire expression is divided.

Solving Equations

Solving a quadratic equation with the quadratic formula involves a few systematic steps:First, identify the coefficients \(a\), \(b\), and \(c\) from your quadratic equation. For example, in \(x^2 - 3x + 1 = 0\), you have \(a = 1\), \(b = -3\), and \(c = 1\).

• Use these coefficients to lay the groundwork for applying the quadratic formula.
• Substitute the values into the quadratic formula to set up the expression that will yield the roots.

Next, take care to simplify the expression under the square root, known as the discriminant \(b^2 - 4ac\). This step determines real and complex roots.

• If the discriminant is positive, there are two distinct real roots.
• If it's zero, there's one real double root.
• If it's negative, the roots are complex and not real numbers.

Finally, solve for the root values by calculating both the positive and negative versions of the formula's \(\pm\) component. Check your solutions by plugging them back into the original equation to ensure they satisfy it.

Roots of Equations

The roots of an equation are the values of \(x\) that satisfy the equation, meaning they make the equation true. For quadratic equations, the roots are where the parabola described by the equation intersects the x-axis. In our example, \(x^2 - 3x + 1 = 0\), solving it with the quadratic formula yielded two roots:

• \(x_1 = \frac{3 + \sqrt{5}}{2}\)
• \(x_2 = \frac{3 - \sqrt{5}}{2}\)

These roots are important for describing the solutions to the equation and can offer insights into the behavior of the function. The nature of the roots—whether real or complex—depends heavily on the discriminant \(b^2-4ac\):

• If positive, the parabola crosses the x-axis twice, indicating two real solutions.
• If zero, it touches the x-axis at one point, indicating one repeated real solution.
• If negative, the parabola does not touch the x-axis, indicating two complex solutions.

Understanding these aspects helps you predict and analyze quadratic functions and their graphs effectively.