Question

If
$\alpha \: and \: \beta \: are \: the \: roots \: of \: the \: \\ quadratic \: equation \\ \: {x}^{2} - 10x + 2 = 0 \: and \: \alpha > \beta \\ find \\ \frac{1}{ \beta } - \frac{1}{ \alpha } \\ { \alpha }^{3} - { \beta }^{3}$
Thanks
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Answer 1

Answer:

☞Given:

$\alpha \: and \: \beta \: are \: the \: roots \: of \: the \: \\ quadratic \: equation \\ \: {x}^{2} - 10x + 2 = 0 \: and \: \alpha > \beta \\</strong></p><h3><strong>☞</strong><strong>To</strong><strong> Find</strong><strong>:</strong></h3><p><strong>frac{1}{ \beta } - \frac{1}{ \alpha } \\ { \alpha }^{3} - { \beta }^{3}$

☞Solution:

$\alpha \: and \beta \: are \: roots \: of \: \\ {x}^{2} - 5x + 6 = 0 \\ (x - 3)(x - 2) = 0 \\ x = 3 \: and \: 2 \\ = > \alpha + \beta = 5 \\ and \: \alpha - \beta = 1$

$The \: equation \: having \: \\ \alpha + \beta \: and \: \alpha - \beta   as \: roots \: is$

${x}^{2} - (sum \: of \: the \: roots)x + \\ (product \: of \: the \: roots) = 0$

${x}^{2} - (5 + 1) x +( 5 \times 1) = 0$

$= > {x}^{2} - 6x + 5 = 0 \\ \: is \: the \: required \: equation$

◉LET'S EXPLORE MORE

✯What is Quadratic Equation?

Quadratic equations are the polynomial equations of degree 2 in one variable of type f(x) = ax2 + bx + c where a, b, c, ∈ R and a ≠ 0. It is the general form of a quadratic equation where ‘a’ is called the leading coefficient and ‘c’ is called the absolute term of f (x). The values of x satisfying the quadratic equation are the roots of the quadratic equation (α,β).

✯Quadratic Equation Formula

The solution or roots of a quadratic equation are given by the quadratic formula:

• (α, β) = [-b ± √(b2 – 4ac)]/2ac

—☆Formulas for Solving Quadratic Equations

1. The roots of the quadratic equation: x = (-b ± √D)/2a, where D = b2 – 4ac

2. Nature of roots:

• D > 0, roots are real and distinct (unequal)
• D = 0, roots are real and equal (coincident)
• D < 0, roots are imaginary and unequal

3. The roots (α + iβ), (α – iβ) are the conjugate pair of each other.

4. Sum and Product of roots: If α and β are the roots of a quadratic equation, then

• S = α+β= -b/a = coefficient of x/coefficient of x2
• P = αβ = c/a = constant term/coefficient of x2

5. Quadratic equation in the form of roots: x2 – (α+β)x + (αβ) = 0

6. The quadratic equations a1x2 + b1x + c1 = 0 and a2x2 + b2x + c2 = 0 have;

• One common root if (b1c2 – b2c1)/(c1a2 – c2a1) = (c1a2 – c2a1)/(a1b2 – a2b1)
• Both roots common if a1/a2 = b1/b2 = c1/c2

7. In quadratic equation ax2 + bx + c = 0 or [(x + b/2a)2 – D/4a2]

• If a > 0, minimum value = 4ac – b2/4a at x = -b/2a.
• If a < 0, maximum value 4ac – b2/4a at x= -b/2a.

8. If α, β, γ are roots of cubic equation ax3 + bx2 + cx + d = 0, then, α + β + γ = -b/a, αβ + βγ + λα = c/a, and αβγ = -d/a

9. A quadratic equation becomes an identity (a, b, c = 0) if the equation is satisfied by more than two numbers i.e. having more than two roots or solutions either real or complex

✯Roots of Quadratic Equation

The values of variables satisfying the given quadratic equation are called its roots. In other words, x = α is a root of the quadratic equation f(x), if f(α) = 0.

• The real roots of an equation f(x) = 0 are the x-coordinates of the points where the curve y = f(x) intersect the x-axis.

• One of the roots of the quadratic equation is zero and the other is -b/a if c = 0
• Both the roots are zero if b = c = 0
• The roots are reciprocal to each other if a = c