Problem 20 - 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 value of b in the given quadratic equation is \(b = 2\).

Step 1: Identify the coefficients

In the given quadratic equation, \(b^2 - 4b + 4 = 0\), the coefficients are: $$ A = 1, B = -4, C = 4 $$

Step 2: Apply the quadratic formula

Now, we will substitute the values of A, B, and C into the quadratic formula: $$ b = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \times 1 \times 4}}{2 \times 1} $$

Step 3: Simplify the expression

Simplify the expression inside and outside the square root: $$ b = \frac{4 \pm \sqrt{16 - 16}}{2} $$ $$ b = \frac{4 \pm \sqrt{0}}{2} $$

Step 4: Calculate the values of b

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

Key Concepts

Quadratic Formula

The quadratic formula is one of the key methods for solving quadratic equations, which are equations of the form \( ax^2 + bx + c = 0 \). This formula provides the solutions to any quadratic equation and is expressed as:

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

This powerful formula can find both real and complex solutions, depending on the values of the coefficients \( a \), \( b \), and \( c \). The part of the formula under the square root, \( b^2 - 4ac \), is known as the discriminant. The value of the discriminant determines the nature of the roots:

• If \(b^2 - 4ac > 0\), there are two distinct real roots.
• If \(b^2 - 4ac = 0\), there is exactly one real root, often referred to as a repeated or double root.
• If \(b^2 - 4ac < 0\), the roots are complex and are not real numbers.

Using the quadratic formula, we can systematically evaluate any quadratic equation and find its solutions, providing clear and precise results.

Solving Quadratic Equations

Solving quadratic equations is a fundamental skill in algebra. There are several methods to solve these equations, including factoring, completing the square, and using the quadratic formula.
In the specific problem \(b^2 - 4b + 4 = 0\), the quadratic formula is employed, which is particularly useful when factoring is not straightforward or when dealing with complex numbers. Here's a brief outline of this process:

• Identify the coefficients: Determine the values of \( a \), \( b \), and \( c \) from the equation.
• Apply the quadratic formula: Substitute these values into the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
• Simplify: Calculate the discriminant \( b^2 - 4ac \) and simplify under the square root to see the nature of the roots.
• Calculate the values: Solve for \( x \) by computing the results for both the plus and minus cases in the formula, ensuring all calculations are simplified.

Solving quadratic equations with precision helps in numerous mathematical situations, from simple algebra tasks to more advanced calculus problems. The quadratic formula provides a universal approach, fitting even the most complex equations.

Algebraic Coefficients

In quadratic equations, the coefficients play a crucial role in determining the equation's characteristics and solutions. The standard form of a quadratic equation is \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are known as the algebraic coefficients.

• Coefficient \( a \): This is the coefficient of \( x^2 \) and is never zero for quadratic equations. It indicates the parabola's width and direction; a positive \( a \) opens upwards, while a negative \( a \) opens downwards.
• Coefficient \( b \): This affects the position of the parabola along the x-axis and interacts with \( a \) to determine its vertex.
• Coefficient \( c \): This represents the y-intercept of the equation, which is the point where the parabola crosses the y-axis.

Understanding these coefficients allows for a deeper insight into the quadratic equation at hand, facilitating the prediction of its graph shape and the type of solutions it might possess. Especially when using the quadratic formula, accurately identifying and applying these values is pivotal for solving the equation correctly.