Problem 110 - Elementary Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify coefficients

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

Step 2: Write the Quadratic Formula

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

Step 3: Substitute coefficients into the Quadratic Formula

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

Step 4: Simplify inside the square root

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

Step 5: Simplify the square root

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

Step 6: Simplify the expression

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

Step 7: State the solutions

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

Key Concepts

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’s 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.