Roots of a Quadratic Equation

IITM Quiz 1

3️⃣ Roots of a Quadratic Equation

Question: If $ \alpha $ and $ \beta $ are the roots of $ x^2 + 4x + 1 = 0 $, then the equation whose roots are $ \alpha^2 $ and $ \beta^2 $ is:

Step-by-Step Solution with Formulas ✨

Let’s find the quadratic equation whose roots are $ \alpha^2 $ and $ \beta^2 $, given that $ \alpha $ and $ \beta $ are the roots of $ x^2 + 4x + 1 = 0 $.

1. Identify the Sums and Products of the Original Roots

For a quadratic equation $ x^2 + px + q = 0 $:

Sum of roots: $ \alpha + \beta = -p $
Product of roots: $ \alpha \beta = q $

So, for $ x^2 + 4x + 1 = 0 $:

$ \alpha + \beta = -4 $
$ \alpha \beta = 1 $

2. Find the Sum of the Squares of the Roots

Use the formula:

$$ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta $$

Plug in the values:

$$ \alpha^2 + \beta^2 = (-4)^2 - 2 \times 1 = 16 - 2 = 14 $$

3. Find the Product of the Squares of the Roots

Use the formula:

$$ \alpha^2 \beta^2 = (\alpha \beta)^2 $$

Plug in the value:

$$ \alpha^2 \beta^2 = (1)^2 = 1 $$

4. Form the New Quadratic Equation

A quadratic equation with roots $ r_1 $ and $ r_2 $ is:

$$ x^2 - (r_1 + r_2)x + (r_1 r_2) = 0 $$

So, for roots $ \alpha^2 $ and $ \beta^2 $:

$$ x^2 - (\alpha^2 + \beta^2)x + (\alpha^2 \beta^2) = 0 $$

Substitute the calculated values:

$$ x^2 - 14x + 1 = 0 $$

5. Final Answer

The required quadratic equation whose roots are $ \alpha^2 $ and $ \beta^2 $ is:

$$ \boxed{x^2 - 14x + 1 = 0}

Summary Table

Step	Formula Used	Value
Sum of roots	$ \alpha + \beta = -p $	$ -4 $
Product of roots	$ \alpha \beta = q $	$ 1 $
Sum of squares	$ (\alpha + \beta)^2 - 2\alpha\beta $	$ 14 $
Product of squares	$ (\alpha\beta)^2 $	$ 1 $
New equation	$ x^2 - (sum)x + (product) = 0 $	$ x^2 - 14x + 1 = 0 $

That’s how you find the new quadratic equation step by step, using formulas at every stage! 🚀

How to Find the Sum and Product of Roots in a Quadratic Equation 🧮

Let’s break down how the sum and product of roots are found for a quadratic equation like $ x^2 + 4x + 1 = 0 $.

1. General Formula for a Quadratic Equation

A quadratic equation is usually written as:

$$ ax^2 + bx + c = 0 $$

The roots (solutions) are usually called $ \alpha $ and $ \beta $.

2. Formulas for Sum and Product of Roots

For the equation $ ax^2 + bx + c = 0 $:

Sum of roots:

$$ \alpha + \beta = -\frac{b}{a} $$

Product of roots:

$$ \alpha \beta = \frac{c}{a} $$

3. Apply to the Given Equation

Given: $ x^2 + 4x + 1 = 0 $

Here, $ a = 1 $, $ b = 4 $, $ c = 1 $

So:

Sum of roots:

$$ \alpha + \beta = -\frac{4}{1} = -4 $$

Product of roots:

$$ \alpha \beta = \frac{1}{1} = 1 $$

4. Summary Table

Formula	Calculation	Result
$ \alpha + \beta = -\frac{b}{a} $	$ -\frac{4}{1} $	$ -4 $
$ \alpha \beta = \frac{c}{a} $	$ \frac{1}{1} $	$ 1 $

5. Emoji Recap!

✍️ Sum of roots: Just take the negative of the coefficient of $ x $ and divide by the coefficient of $ x^2 $!
✍️ Product of roots: Just divide the constant term by the coefficient of $ x^2 $!

That’s how the sum and product of roots are found for any quadratic equation! 🚀

Relations and Sets Sum of Squared Errors (SSE) for Best-Fit Line