Problem 44 - Elementary Algebra

Question

Use the quadratic formula to solve. $$ t 2-44 t+484=0 $$

Step-by-Step Solution

Answer

The solution is \( t = 22 \).

Step 1: Identify the coefficients

For the quadratic equation in the form \( ax^2 + bx + c = 0 \), identify the coefficients \( a \), \( b \), and \( c \). In this equation, \( a = 1 \), \( b = -44 \), and \( c = 484 \).

Step 2: Write the quadratic formula

The quadratic formula for finding the roots of an equation \( ax^2 + bx + c = 0 \) is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \].

Step 3: Calculate the discriminant

The discriminant \( \Delta \) is calculated as \( b^2 - 4ac \). Substitute the values of \( b \), \( a \), and \( c \): \( \Delta = (-44)^2 - 4(1)(484) \). Calculate this to get \( \Delta = 1936 - 1936 = 0 \).

Step 4: Apply the quadratic formula

Since the discriminant is 0, there will be one repeated real root. Substitute the values into the quadratic formula: \[ t = \frac{-(-44) \pm \sqrt{0}}{2 \cdot 1} \]. This simplifies to \( t = \frac{44}{2} \), which gives \( t = 22 \).

Step 5: Conclude with the solution

Since there is only one root due to the discriminant being zero, there is exactly one solution: \( t = 22 \).

Key Concepts

Understanding the Quadratic Equation

A quadratic equation is any equation that can be rearranged in the standard form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. Understanding the role of these coefficients is essential:

• \( a \) determines the parabola's direction (upward if positive, downward if negative).
• \( b \) controls the parabola's vertex’s position horizontally along the x-axis.
• \( c \) represents the y-intercept, where the graph intersects the y-axis.

A quadratic equation forms a U-shaped curve called a parabola. The goal when solving the quadratic equation is to find the points (roots) where this parabola crosses the x-axis, if it does cross it. You might realize that not all parabolas cut the x-axis, which leads us to our next important concept: the discriminant.

Role of the Discriminant in Solving Quadratics

The discriminant is a component of the quadratic formula, calculated as \( b^2 - 4ac \). It plays a crucial role in determining the nature of the roots of a quadratic equation:

• If the discriminant is positive, the equation has two distinct real roots.
• If the discriminant is zero, the equation has one repeated real root.
• If the discriminant is negative, the equation has two complex roots, with no real solutions.

For the equation in the exercise \( t^2 - 44t + 484 = 0 \), the discriminant was found to be zero, indicating that there is exactly one real root. This means the parabola just touches the x-axis at one point, exemplifying a perfect square trinomial.

The Roots of an Equation

The roots of a quadratic equation are the values of \( x \) that satisfy the equation. These can be referred to as solutions or zeroes of the equation.

• The roots are found using the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
• The symbol \( \pm \) in the formula signifies that there are generally two solutions, one by adding the square root and one by subtracting it.
• When the discriminant is zero, the \( \pm \sqrt{0} \) simplifies the expression, showing that there is only one solution.

In our example, using the quadratic formula yielded a single root \( t = 22 \). Because the discriminant was zero, the parabola touches the x-axis only at this root, thus confirming the single solution is indeed the vertex of the parabola at this point on the graph.