Problem 4 - Algebra 2

Question

Find the exact solutions by using the Quadratic Formula. \(x^{2}+6 x+9=0\)

Step-by-Step Solution

Answer

The solution is \(x = -3\).

Step 1: Identify Coefficients

In the quadratic equation of the form \(ax^2 + bx + c = 0\), identify \(a\), \(b\), and \(c\). Here, \(a = 1\), \(b = 6\), and \(c = 9\).

Step 2: Write the Quadratic Formula

The quadratic formula is given by \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

Step 3: Calculate the Discriminant

Substitute \(b\), \(a\), and \(c\) into \(b^2 - 4ac\) to find the discriminant. Here, it is \(6^2 - 4 \times 1 \times 9 = 36 - 36 = 0\).

Step 4: Interpret the Discriminant

Since the discriminant is 0, this means the quadratic equation has exactly one real solution, which is a repeated root.

Step 5: Solve for the Repeated Root

Substitute the values and the discriminant back into the quadratic formula: \(x = \frac{-6 \pm \sqrt{0}}{2 \times 1} = \frac{-6}{2} = -3\).

Key Concepts

Quadratic Equation

A quadratic equation is a polynomial equation of the second degree. It has the standard form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable. Quadratic equations are fundamental in algebra and appear frequently in various real-world scenarios, such as physics, engineering, and finance.
To solve quadratic equations, one can use several methods, including:

• Factoring
• Completing the square
• The quadratic formula

Each method has its own advantages, but the quadratic formula is particularly useful because it always works, no matter how complex the equation may seem.

Discriminant

The discriminant is a specific part of the quadratic formula that helps determine the nature of the roots of a quadratic equation. It is found under the square root in the formula: \( b^2 - 4ac \).
The discriminant gives us critical information about the solutions:

• If \( b^2 - 4ac > 0 \), there are two distinct real solutions.
• If \( b^2 - 4ac = 0 \), there is exactly one real solution, called a repeated root.
• If \( b^2 - 4ac < 0 \), there are no real solutions, only complex ones.

This knowledge allows us to quickly assess the types of solutions without fully solving the equation.

Real Solutions

Real solutions refer to the solutions of a quadratic equation that are real numbers. Depending on the value of the discriminant, a quadratic equation can have one, two, or no real solutions.

• Two distinct real solutions occur when the discriminant is positive.
• A single real solution, known as a repeated root, happens when the discriminant is zero.
• If there are no real solutions, this is typically because the discriminant is negative, resulting in complex solutions instead.

Understanding whether a quadratic equation will yield real solutions is essential for interpreting the results in a real-world context, where real numbers usually represent measurable quantities.

Repeated Root

A repeated root happens when a quadratic equation has a discriminant of zero. This means the equation has exactly one real solution. The root is 'repeated' because both possible solutions from the quadratic formula are the same.
The general form of the solution in this case will be: \[ x = \frac{-b}{2a} \] For the equation \( x^2 + 6x + 9 = 0 \), the solution is \( x = -3 \).
Repeated roots occur when a quadratic equation is a perfect square trinomial, like \((x + 3)^2 = 0\). In this form, the only solution is \( x = -3 \), which occurs twice but results in a single, unique solution in practice.