Problem 123 - Elementary Algebra

Question

Solve by using the Quadratic Formula. \(4 r^{2}+3 r-5=0\)

Step-by-Step Solution

Answer

The solutions are \( r_1 = \frac{{-3 + \sqrt{89}}}{8} \) and \( r_2 = \frac{{-3 - \sqrt{89}}}{8} \).

Step 1 - Identify coefficients

The quadratic equation is in the form of

Step 2 - Plug coefficients into the Quadratic Formula

The Quadratic Formula is given by \( r = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \). Substitute the coefficients: \( a = 4 \), \( b = 3 \), \( c = -5 \).

Step 3 - Calculate the discriminant

The discriminant is calculated as \( b^2 - 4ac \): \( 3^2 - 4 \times 4 \times (-5) = 9 + 80 = 89 \) .

Step 4 - Compute the solutions

Now, substitute the discriminant back into the Quadratic Formula: \( r = \frac{{-3 \pm \sqrt{89}}}{8} \). This yields two solutions: \( r_1 = \frac{{-3 + \sqrt{89}}}{8} \), \( r_2 = \frac{{-3 - \sqrt{89}}}{8} \).

Key Concepts

Quadratic Equations

Quadratic equations are mathematical expressions set to zero, having the standard form:

• \[ ax^2 + bx + c = 0 \]

Here, \(a\), \(b\), and \(c\) are coefficients, and \(x\) represents the variable. The term \(ax^2\) makes it quadratic because 'quadratic' refers to the square of the variable.

Quadratic equations are used to model various real-world scenarios like projectile motion, areas, and economics. Solving these equations requires techniques such as factoring, completing the square, or using the Quadratic Formula.

In the given exercise, the equation is:

• \[ 4r^2 + 3r - 5 = 0 \]

This fits into the standard form, where:\(a = 4\), \(b = 3\), and \(c = -5\).

Discriminant

The discriminant is a key part of the Quadratic Formula and provides useful information about the nature of the solutions.

The discriminant is calculated as:

• \[ D = b^2 - 4ac \]

Here, \(D\) helps us understand the types of roots the quadratic equation will have:

• If \(D > 0\): There are two distinct real solutions.
• If \(D = 0\): There is exactly one real solution (a repeated root).
• If \(D < 0\): There are no real solutions, but two complex solutions.

For the exercise, plugging in the values gives us:

• \[ D = 3^2 - 4(4)(-5) = 9 + 80 = 89 \]

Since \(D = 89\), which is greater than 0, we can expect two distinct real solutions.

Solving Quadratic Equations

The Quadratic Formula is a universal method for solving any quadratic equation. The formula is given by:

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

Let's solve the given equation step-by-step:

1. Identify the coefficients \(a\), \(b\), and \(c\):
\[ a = 4, \ b = 3, \ c = -5 \]
2. Calculate the discriminant:

• \( D = 89 \)

3. Substitute \(a\), \(b\), and \(D\) into the formula:

• \[ r = \frac{{-3 \pm \sqrt{89}}}{8} \]

This computation gives us two solutions:

• \[ r_1 = \frac{{-3 + \sqrt{89}}}{8} \]
• \[ r_2 = \frac{{-3 - \sqrt{89}}}{8} \]

This shows how the quadratic formula can solve equations that might be difficult to factor or simplify otherwise.