Question

अल्फा प्लस बीटा इज इक्वल टू थ्री एंड अल्फा बीटा इज इक्वल टू माइनस 4 find quadratic equation
$\alpha + \beta = 3 \: and \: \alpha \beta = 4find \: qadratic \: equetion$
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Answer 1

Question:

If $\displaystyle{\sf\:\alpha\:+\:\beta\:=\:3\:\&\:\alpha\:.\:\beta\:=\:-\:4}$, then find the quadratic equation.

Answer:

The required quadratic equation is $\displaystyle{\boxed{\red{\sf\:x^2\:-\:3x\:-\:4\:=\:0}}}$.

Step-by-step-explanation:

We have given the sum and product of the roots of the quadratic equation.

We have to find the quadratic equation.

Now,

$\displaystyle{\bullet\sf\:Sum\:of\:roots\:=\:\alpha\:+\:\beta\:=\:3}$

$\displaystyle{\bullet\sf\:Product\:of\:roots\:=\:\alpha\:.\:\beta\:=\:-\:4}$

Now, we know that,

The required quadratic equation is in the form

$\displaystyle{\pink{\sf\:x^2\:-\:(\:\alpha\:+\:\beta\:)\:x\:+\:(\:\alpha\:.\:\beta\:)\:=\:0}}$

$\displaystyle{\implies\sf\:x^2\:-\:(\:3\:)\:x\:+\:(\:-\:4\:)\:=\:0}$

$\displaystyle{\implies\sf\:x^2\:-\:3x\:-\:4\:=\:0}$

The required quadratic equation is $\displaystyle{\boxed{\red{\sf\:x^2\:-\:3x\:-\:4\:=\:0}}}$.

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Additional Information:

1. Quadratic Equation:

An equation with degree "2" is called as quadratic equation.

The general form of a quadratic equation is ax² + bx + c = 0.

Where, a, b, c are real numbers and a ≠ 0.

2. Roots of Quadratic Equation:

The roots are the values of the variable used in the equation for which the LHS and RHS of the equation becomes equal.

3. Methods of solving quadratic equation:

There are mainly three methods to solve a quadratic equation.

A. Factorization method

B. Completing square method

C. Formula method

4. Formula to solve quadratic equation:

$\displaystyle{\boxed{\red{\sf\:x\:=\:\dfrac{-\:b\:\pm\:\sqrt{b^2\:-\:4ac}}{2a}}}}$

5. Forming a quadratic equation when the roots are given:

If we have the roots of a quadratic equation, we can form the quadratic equation.

If $\displaystyle{\sf\:\alpha\:\&\:\beta}$ are the roots of a quadratic equation, then the quadratic equation is in the form:

$\displaystyle{\sf\:x^2\:-\:(\:Sum\:of\:roots\:)\:x\:+\:(\:Product\:of\:roots\:)\:=\:0}$

$\displaystyle{\therefore\:\boxed{\red{\sf\:x^2\:-\:(\:\alpha\:+\:\beta\:)\:x\:+\:(\:\alpha\:.\:\beta\:)\:=\:0}}}$

Answer 2

Answer:

✯ Given :-

◕ α + β = 3

◕ αβ = - 4

✯ To Find :-

◐ What is the qradratic equation.

✯ Solution :-

⋆ Given :-

• Sum of the roots = 3
• Product of the roots = - 4

↦ To find qradratic equation we know that,

$\boxed{\bold{\small{✭\:x²\: -\: (Sum\: of\: the\: roots)\: x\: +\: (Product\: of\: the\: roots)\: =\: 0\:✭}}}$

➙ Required equation :-

➣ x² - (Sum of the roots) x + (Product of the roots) = 0

➣ x² - (α + β) x + (αβ) = 0

» By putting the given value we get,

⇒ x² - (3) x + (- 4) = 0

➠ x² - 3x - 4 = 0

$\therefore$ The required qradratic equation is $\small\purple{\underline{{\boxed{\textbf{x² - 3x - 4 = 0}}}}}$

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❖ Extra Information ❖

✪ Quadratic equation ✪

» The name Quadratic comes from the words "quad" meaning square, because the variables gets squared (like x²).

» An equation of the form ax² + bx + c = 0 is known as a quadratic equation. (Here, x is the variables and a,b and c is a constant where a≠0 )

✮ Important Formula ✮

➞ The sums of the roots α and β of the qradratic equation are :-

✶ α + β = - $\dfrac{b}{c}$ ✶

➞ The product of the roots α and β of the qradratic equation are :-

✶ αβ = $\dfrac{c}{a}$ ✶

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