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Chapter 10: Problem 110

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

Short Answer

Expert verified
The solutions are \( b = \frac{-1 + \sqrt{21}}{5} \) and \( b = \frac{-1 - \sqrt{21}}{5} \).

Step by step solution

01

Identify coefficients

In the quadratic equation, identify the coefficients: \(a = 5\), \(b = 2\), and \(c = -4\).
02

Write the Quadratic Formula

Recall the Quadratic Formula: \[ b = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
03

Substitute coefficients into the Quadratic Formula

Substitute \(a = 5\), \(b = 2\), and \(c = -4\) into the Quadratic Formula: \[ b = \frac{-2 \pm \sqrt{2^2 - 4(5)(-4)}}{2(5)} \]
04

Simplify inside the square root

Calculate inside the square root: \[ 2^2 - 4(5)(-4) = 4 + 80 = 84 \]So our formula now is \[ b = \frac{-2 \pm \sqrt{84}}{10} \]
05

Simplify the square root

Simplify \( \sqrt{84} \): \[ \sqrt{84} = 2\sqrt{21} \]Thus, the formula becomes: \[ b = \frac{-2 \pm 2\sqrt{21}}{10} \]
06

Simplify the expression

Factor out common terms: \[ b = \frac{2(-1 \pm \sqrt{21})}{10} = \frac{-1 \pm \sqrt{21}}{5} \]
07

State the solutions

The solutions for the quadratic equation are: \[ b = \frac{-1 + \sqrt{21}}{5} \]and\[ b = \frac{-1 - \sqrt{21}}{5} \]

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Key Concepts

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

Identifying Coefficients
To solve a quadratic equation using the quadratic formula, you first need to identify the coefficients. The general form of a quadratic equation is \(ax^2 + bx + c = 0\). Here, \(a\), \(b\), and \(c\) are the coefficients. In the given equation, \(5b^2 + 2b - 4 = 0\), the coefficients are:
  • \(a = 5\)
  • \(b = 2\)
  • \(c = -4\)
Identifying these values correctly is crucial for accurately applying the quadratic formula. It's like finding the correct ingredients for your recipe before you start cooking!
Solving Quadratic Equations
With the coefficients identified, we can now use the quadratic formula to find the solutions to the equation. The quadratic formula is given by:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
This formula might seem complicated at first, but it鈥檚 straightforward once you understand it. By substituting the coefficients \(a = 5\), \(b = 2\), and \(c = -4\) into the formula, we get:
\[x = \frac{-2 \pm \sqrt{2^2 - 4(5)(-4)}}{2(5)}\]
The next steps involve simplifying the expression under the square root (discriminant) and then simplifying the entire expression.
Simplifying Square Roots
Simplifying the square root is a key part of solving the quadratic equation. For our example, the discriminant inside the square root is:
\[2^2 - 4(5)(-4)\]
First, calculate the product inside the parentheses:
\[2^2 = 4\] \[-4(5)(-4) = 80\]
Add these values together:
\[4 + 80 = 84\]
This means we have \(\sqrt{84}\). To simplify \(\sqrt{84}\), we break it down into factors:
\[\sqrt{84} = \sqrt{4 \cdot 21} = 2\sqrt{21}\]
Always try to simplify the square root to make further calculations easier.
Mathematical Expressions
Finally, we simplify the entire mathematical expression to find the solutions. Substitute \(\sqrt{84}\) with \(2\sqrt{21}\) and simplify:
\[x = \frac{-2 \pm 2\sqrt{21}}{10}\]
Factor out the common term (2):
\[x = \frac{2(-1 \pm \sqrt{21})}{10} = \frac{-1 \pm \sqrt{21}}{5}\]
This gives us the solutions to the quadratic equation:
\[x = \frac{-1 + \sqrt{21}}{5}\]
and
\[x = \frac{-1 - \sqrt{21}}{5}\]
Breaking down the expressions step-by-step helps in better understanding and solving quadratic equations accurately.

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Most popular questions from this chapter

Rafi is designing a rectangular playground to have an area of 320 square feet. He wants one side of the playground to be four feet longer than the other side. Solve the equation \(p^{2}+4 p=320\) for \(p,\) the length of one side of the playground. What is the length of the other side?

The hypotenuse of a right triangle is \(10 \mathrm{~cm}\) long. One of the triangle's legs is three times the length of the other leg. Find the lengths of the three sides of the triangle. Round to the nearest tenth.

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

Recognize the Graph of a Quadratic Equation in Two Variables. $$ y=x^{2}+3 $$

Solve by using the Quadratic Formula. \(6 z^{2}-9 z+1=0\)
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