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Chapter 10: Problem 123

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

Short Answer

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

Step by step solution

01

- Identify coefficients

The quadratic equation is in the form of
02

- Plug coefficients into the Quadratic Formula

The Quadratic Formula is given by \( r = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \). Substitute the coefficients: \( a = 4 \), \( b = 3 \), \( c = -5 \).
03

- Calculate the discriminant

The discriminant is calculated as \( b^2 - 4ac \): \( 3^2 - 4 \times 4 \times (-5) = 9 + 80 = 89 \) .
04

- Compute the solutions

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

An architect is designing a hotel lobby. She wants to have a triangular window looking out to an atrium, with the width of the window 6 feet more than the height. Due to energy restrictions, the area of the window must be 140 square feet. Solve the equation \(\frac{1}{2} h^{2}+3 h=140\) for \(h,\) the height of the window.

Make up a problem involving the product of two consecutive even integers. Start by choosing two consecutive even integers. (a) What are your integers? (b) What is the product of your integers? (c) Solve the equation \(n(n+2)=p,\) where \(p\) is the product you found in part (b). (d) Did you get the numbers you started with?

In the following exercises, determine if the parabola opens up or down. $$ y=6 x^{2}+2 x+3 $$

Solve the equation \(12 y^{2}+23 y=24\) (a) by completing the square (b) using the Quadratic Formula ( Which method do you prefer? Why?

Solve by using the Quadratic Formula. \(25 q^{2}+30 q+9=0\)
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