Problem 25 - Elementary Algebra

Question

Use the quadratic formula to solve the following. $$t 2-4 t-1=0$$

Step-by-Step Solution

Answer

The solutions are \( t = 2 + \sqrt{5} \) and \( t = 2 - \sqrt{5} \).

Step 1: Identify the coefficients

The given quadratic equation is in the standard form of a quadratic equation, which is \( ax^2 + bx + c = 0 \). Here, \( t^2 - 4t - 1 = 0 \). The coefficients are \( a = 1 \), \( b = -4 \), and \( c = -1 \).

Step 2: Write down the quadratic formula

The quadratic formula to solve the equation \( ax^2 + bx + c = 0 \) is: \[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] We will use this formula to find the solution for the given equation.

Step 3: Calculate the discriminant

The discriminant of the quadratic equation is given by \( b^2 - 4ac \). Substitute \( a = 1 \), \( b = -4 \), and \( c = -1 \) into the discriminant formula:\( (-4)^2 - 4(1)(-1) = 16 + 4 = 20 \)

Step 4: Substitute into the quadratic formula

Substitute \( a = 1 \), \( b = -4 \), and the discriminant value into the quadratic formula:\[ t = \frac{-(-4) \pm \sqrt{20}}{2(1)} = \frac{4 \pm \sqrt{20}}{2} \]

Step 5: Simplify the expression

The expression \( \sqrt{20} \) can be simplified to \( 2\sqrt{5} \). Thus the solution becomes:\[ t = \frac{4 \pm 2\sqrt{5}}{2} \] Simplify further by dividing each term by 2:\[ t = 2 \pm \sqrt{5} \]

Step 6: Interpret the solution

The solutions to the quadratic equation are \( t = 2 + \sqrt{5} \) and \( t = 2 - \sqrt{5} \). These are the two possible values of \( t \) that satisfy the original equation.

Key Concepts

Quadratic Formula

In mathematics, the quadratic formula is a powerful tool used to find the roots of quadratic equations, which are equations of the form \( ax^2 + bx + c = 0 \). For any quadratic equation, the formula to find the solutions is given by:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\].This formula is particularly useful because it can be used to solve any quadratic equation, regardless of the values of \( a \), \( b \), and \( c \). It calculates the roots by determining the values of \( x \) that satisfy the equation.

Coefficients

Coefficients are the numerical factors that multiply the variable terms in a polynomial equation. In the standard form of a quadratic equation, \( ax^2 + bx + c = 0 \), the coefficients are:

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

Understanding the coefficients is crucial because they directly influence the shape and position of the parabola represented by the quadratic equation. They also play a critical role in calculating the roots using the quadratic formula.

Discriminant

The discriminant is an important component of the quadratic formula and is given by \( b^2 - 4ac \). It provides valuable information about the nature of the roots of the quadratic equation.

• If the discriminant is positive, the quadratic equation has two distinct real roots.
• If the discriminant is zero, there is exactly one real root, known as a repeated or double root.
• If the discriminant is negative, the quadratic equation has two complex roots.

The value of the discriminant helps us anticipate the type of solutions we can expect before even solving the equation.

Solution of Quadratic Equations

Solving quadratic equations can be accomplished through several methods, but the quadratic formula method is one of the most universally applicable. Given the completed formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), the solutions are found by carefully:

• Identifying the coefficients \( a \), \( b \), and \( c \).
• Calculating the discriminant \( b^2 - 4ac \).
• Substituting these values into the quadratic formula.
• Simplifying the resulting expressions to find the exact values of \( x \).

In the given context of the equation \( t^2 - 4t - 1 = 0 \), the calculated solutions \( t = 2 + \sqrt{5} \) and \( t = 2 - \sqrt{5} \) show how the formula is applied step-by-step to find the roots.