Problem 13 - College Algebra with Modeling and Visualization

Question

Exercises \(1-28:\) Solve the quadratic equation. Check your answers for Exercises \(1-12\). $$ (t+3)^{2}=5 $$

Step-by-Step Solution

Answer

The solutions are \( t = -3 + \sqrt{5} \) and \( t = -3 - \sqrt{5} \).

Step 1: Expand the equation

Start by expanding the left side of the equation: \((t+3)^2 = 5\) to get \(t^2 + 6t + 9 = 5\).

Step 2: Simplify the equation

Subtract 5 from both sides to simplify:\[ t^2 + 6t + 9 - 5 = 0 \]This simplifies to:\[ t^2 + 6t + 4 = 0 \]

Step 3: Apply the quadratic formula

To solve \( t^2 + 6t + 4 = 0 \), use the quadratic formula \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = 6 \), and \( c = 4 \).

Step 4: Calculate the discriminant

Compute the discriminant: \[ b^2 - 4ac = 6^2 - 4\times 1 \times 4 = 36 - 16 = 20 \]Since the discriminant is positive, there are two real roots.

Step 5: Solve for the roots

Using the quadratic formula, compute the roots:\[ t = \frac{-6 \pm \sqrt{20}}{2} \]Simplify it further:\[ t = \frac{-6 \pm 2\sqrt{5}}{2} \]\[ t = -3 \pm \sqrt{5} \]So, the roots are \( t = -3 + \sqrt{5} \) and \( t = -3 - \sqrt{5} \).

Step 6: Verify the solutions

Substitute \( t = -3 + \sqrt{5} \) and \( t = -3 - \sqrt{5} \) back into the original equation \((t+3)^2 = 5\) to verify each solution. Both should satisfy the equation, confirming the roots are correct.

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. A quadratic equation typically takes the form of \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). The quadratic formula is given by:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This formula provides an explicit solution for \( x \), the unknown variable in the equation. It works by utilizing the coefficients directly from the equation.

• \( a \) is the coefficient of \( x^2 \).
• \( b \) is the coefficient of \( x \).
• \( c \) is the constant term.

Using this formula helps to find the roots of the quadratic equation effectively without the need for graphing or factoring, which can sometimes be complex or time-consuming.

Discriminant

The discriminant is an essential component of the quadratic formula, represented as \( b^2 - 4ac \). It tells us a lot about the roots of a quadratic equation even before solving it completely. The value of the discriminant can help determine the nature of the roots:

• If the discriminant is positive \((b^2 - 4ac > 0)\), there are two distinct real roots.
• If the discriminant equals zero \((b^2 - 4ac = 0)\), there is exactly one real root, or a repeated root.
• If the discriminant is negative \((b^2 - 4ac < 0)\), there are no real roots; instead, the roots are complex numbers.

In this particular exercise, the discriminant was calculated as 20, indicating two distinct real roots exist for the equation.

Real Roots

Real roots are the solutions of a quadratic equation that are real numbers. When you solve a quadratic equation using the quadratic formula, the solutions or roots may be real or complex.
For an equation to have real roots, its discriminant must be non-negative. Both the roots found in the exercise, \( t = -3 + \sqrt{5} \) and \( t = -3 - \sqrt{5} \), are real numbers, meaning they can be plotted on the number line.
Real roots can be:

• Distinct: When they are different from each other, as with two unique solutions.
• Repeated: When both roots are the same.

The real roots provide x-values where the parabola (graph of the quadratic equation) crosses the x-axis.

Solving Equations

Solving equations involves finding the values of the variable that satisfy the equation. In quadratic equations, you're typically looking to find the 'roots,' or solutions, that make the equation equal zero.
Here’s a step-by-step approach illustrated in the given solution:

• Begin by expanding and simplifying the equation if necessary, which leads to the standard quadratic form.
• Use the quadratic formula to find the roots, leveraging the coefficients \( a \), \( b \), and \( c \).
• Calculate the discriminant to understand the nature of the roots before proceeding further.
• Substitute the solutions back into the original equation to verify their correctness. This ensures no errors have been made during calculations.

Properly solving quadratic equations can reveal many useful properties about their graphical representation and behavior.