Problem 14 - Calculus

Question

Find the sum and product of the numbers. The two roots of \(c^{2}-4 c+5=0\)

Step-by-Step Solution

Answer

Sum: 4, Product: 5

Step 1: Identify the Quadratic Equation

The given equation is in the form of a quadratic equation, which is\[c^2 - 4c + 5 = 0\]This equation can be compared to the general quadratic formula \(ax^2 + bx + c = 0\) where \(a = 1\), \(b = -4\), and \(c = 5\).

Step 2: Use the Sum of Roots Formula

The sum of the roots of a quadratic equation \(ax^2 + bx + c = 0\) is given by the formula\[r_1 + r_2 = -\frac{b}{a}\]Substituting \(b = -4\) and \(a = 1\), we have\[r_1 + r_2 = -\frac{-4}{1} = 4\]So, the sum of the roots is 4.

Step 3: Use the Product of Roots Formula

The product of the roots of a quadratic equation \(ax^2 + bx + c = 0\) is given by the formula\[r_1 \times r_2 = \frac{c}{a}\]Substituting \(c = 5\) and \(a = 1\), we have\[r_1 \times r_2 = \frac{5}{1} = 5\]Thus, the product of the roots is 5.

Key Concepts

Sum of the Roots

In the enchanting world of quadratic equations, the sum of the roots holds a special place. When dealing with a quadratic equation of the form \( ax^2 + bx + c = 0 \), you can find the sum of its roots using the easy-peasy formula: \( r_1 + r_2 = -\frac{b}{a} \). Here's the trick, it's like magic! Just grab the coefficient of \( x \) (which we call \( b \)), make it negative, and divide it by \( a \), the coefficient of \( x^2 \).
For example, in our equation \( c^2 - 4c + 5 = 0 \), let's find the sum. With \( a = 1 \) and \( b = -4 \), plug them into the sum formula:
\[ r_1 + r_2 = -\frac{-4}{1} = 4 \]
Voila! The roots joyfully add up to 4.
This method works every time, just like your favorite piece of cake. Give it a try with any quadratic equation, and you'll see why math can be magical.

Product of the Roots

Finding the product of the roots in a quadratic equation is like putting two puzzle pieces together. You use another special formula: \( r_1 \times r_2 = \frac{c}{a} \). Here, \( c \) is the constant term without a hitch.
In our example, where the equation is \( c^2 - 4c + 5 = 0 \), to find the product, you just plug in \( a = 1 \) and \( c = 5 \) into the formula:
\[ r_1 \times r_2 = \frac{5}{1} = 5 \]
There you have it! The roots multiply to give 5.
Doesn’t that remind you of a sweet recipe where mixing two things just gives you the right flavor? This simple method works like a charm to discover the root products, making sure your math meal is delicious!

Quadratic Formula

When quadratic equations wave hello, meeting them with a solution is made easy with the Quadratic Formula. It’s like a bridge connecting you to the right answers, especially when the roots are not that obvious.
The Quadratic Formula is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). It looks grand, but it's super handy. It helps you find the roots (solutions) for the equation \( ax^2 + bx + c = 0 \).
Here’s the play, step by step:

• Calculate \( b^2 - 4ac \), that’s called the discriminant.
• If it’s positive, you have two distinct roots.
• If it’s zero, it’s a perfect square with one root.
• If negative, roots are imaginary.

This formula is like your magic wand for solving quadratics! So next time you stumble upon a tricky equation, just pull out this tool and you'll find the solution with ease. It’s like having hidden superpowers in the world of equations.