Problem 21 - Elementary Algebra

Question

For the following problems, solve the equations using the quadratic formula. $$ b^{2}+4 b+4=0 $$

Step-by-Step Solution

Answer

Answer: The solution to the given quadratic equation is \(x = -2\).

Step 1: Identify the coefficients

In the given equation, \(b^2 + 4b + 4 = 0\), the coefficients are: a = 1 (coefficient of \(x^2\), but not shown), b = 4 (coefficient of x), and c = 4.

Step 2: Apply the quadratic formula

Now that we have identified the coefficients, we can substitute them into the quadratic formula: $$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$ Substitute a = 1, b = 4, and c = 4: $$ x = \frac{-4 \pm \sqrt{4^2 - 4(1)(4)}}{2(1)} $$

Step 3: Simplify the expression

Now let's simplify the expression under the square root and the denominator: $$ x = \frac{-4 \pm \sqrt{16 - 16}}{2} $$ This simplifies to: $$ x = \frac{-4 \pm \sqrt{0}}{2} $$

Step 4: Calculate the roots/solution

Since the square root of 0 is 0, the expression becomes: $$ x = \frac{-4 \pm 0}{2} $$ There is only one solution in this case: $$ x = \frac{-4}{2} $$ So, the solution is: $$ x = -2 $$ The solution to the quadratic equation \(b^2 + 4b + 4 = 0\) using the quadratic formula is \(x = -2\).

Key Concepts

Quadratic Equations

Quadratic equations are mathematical expressions that follow the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( x \) represents the unknown variable. Quadratic equations are prevalent in various areas of mathematics and real-world applications, helping to resolve problems involving area, motion, and even financial forecasting.

When solving a quadratic equation, the goal is to find the values of \( x \) that satisfy the equation. This can be achieved by various methods such as factoring, completing the square, or using the quadratic formula. In particular, the quadratic formula is a powerful tool because it can solve any quadratic equation, even when it is not factorable or easily simplified.

Understanding the structure of a quadratic equation is the first step towards solving them efficiently. The equation typically consists of a squared term \( x^2 \), a linear term \( x \), and a constant term \( c \). These terms together define the parabola shape characteristic of quadratic functions when graphed.

Coefficients

In any quadratic equation, coefficients play a crucial role. The coefficients are the numbers facing the variables in the equation. For quadratic equations, these are typically referred to as \( a \), \( b \), and \( c \).

• \( a \) is the coefficient of \( x^2 \). When \( a = 1 \), this is called a monic quadratic equation.
• \( b \) is the coefficient of \( x \), the linear term. It represents the slope or tilt of the parabola's opening.
• \( c \) is the constant term and represents the y-intercept when the equation is graphed.

In our example, \( b^2 + 4b + 4 = 0 \), the coefficients are identified as: \( a = 1 \), \( b = 4 \), and \( c = 4 \). Understanding how to extract these coefficients from an equation is essential for utilizing the quadratic formula effectively.

Each coefficient gives valuable information about the parabola represented by the quadratic equation. The coefficient \( a \) determines whether the parabola opens upwards or downwards. If \( a > 0 \), it opens upwards. If \( a < 0 \), it opens downwards. The coefficient \( b \) and \( c \) influence the position and symmetry of the parabola, affecting how the equation is solved.

Discriminant

The discriminant is part of the quadratic formula, found under the square root in the expression \( b^2 - 4ac \). It provides critical insight into the nature of the roots of a quadratic equation.

The discriminant helps determine:

• **The number of real roots:** If the discriminant is positive, there are two distinct real roots.
• **The nature of the roots:** If the discriminant equals zero, there is exactly one real root (a repeated root), as seen in the example problem where \( \, b^2 - 4(1)(4) = 0 \, \).
• **The complexity of the roots:** If the discriminant is negative, there are no real roots, just two complex roots.

Understanding the discriminant is crucial as it helps you anticipate the type of solutions you might get without fully solving the equation. In the example \( b^2 + 4b + 4 = 0 \), the discriminant is zero, indicating a single root, \( x = -2 \), obtained from calculating \( \, \frac{-4 \pm 0}{2} \, \), which simplifies neatly into a single value.

Roots of Equations

The roots of a quadratic equation are the solutions for \( x \) that satisfy the equation \( ax^2 + bx + c = 0 \). These roots are where the equation equals zero and are often where the graph of the quadratic intersects the x-axis.

Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), you can find the roots by substituting the coefficients into the formula. The "\( \pm \)" symbol in the formula suggests that there may be two solutions: one for each sign.

In the example problem \( b^2 + 4b + 4 = 0 \), waiting to determine the roots involves solving the expression \( x = \frac{-4 \pm 0}{2} \). Here, the single root is \( x = -2 \).

• **Unique root:** Often occurs when the discriminant is zero, leading to a single intersection at the x-axis.
• **Real and distinct roots:** When the discriminant is positive, indicating two separate x-intercepts.
• **Complex roots:** If the discriminant is negative, indicating no real x-axis intersections.

The roots give important insights into the nature and behavior of the quadratic graph, making them a fundamental concept in understanding quadratic equations.