Question

the quadratic equation whose roots are 1and -1/2​

Answer 1

Given :

• Roots of a quadratic equation are 1 and $\sf{-\dfrac{1}{2}}$

To Find :

• The quadratic equation

Knowledge required :

A quadratic equation (in x) whose roots are α , β is given by

$\\ \star \: {\boxed{\purple{\sf{ x {}^{2} - x( \alpha + \beta) + \alpha \beta }}}}$

Solution :

We have the roots as 1 and $\sf{-\dfrac{1}{2}}$

Sum of the roots is ,

$\\ \sf{\longrightarrow -\dfrac{1}{2}+1}$

$\\ \sf{\longrightarrow\dfrac{-1+2}{2}}$

$\\ \sf{\longrightarrow\dfrac{1}{2}}$

Product of roots is ,

$\\ \sf{\longrightarrow -\dfrac{1}{2}\times 1}$

$\\ \sf{\longrightarrow -\dfrac{1}{2}}$

Now applying the relation ,

$\\ \longrightarrow \sf {x}^{2} - x\bigg( \frac{1}{2} \bigg) + \bigg(- \frac{1}{2} \bigg) = 0$

$\\ \longrightarrow \sf \: {x}^{2} - \frac{x}{2} - \frac{1}{2} = 0$

Multiplying the whole equation with 2 we get ,

$\\ \longrightarrow \underline{\boxed {\pink{\sf {\: 2 {x}^{2} - x - 1 = 0}}}} \: \bigstar$

Hence ,

• The quadratic equation whose roots are 1 and $\sf{-\dfrac{1}{2}}$ is 2x² - x - 1.

Answer 2

Answer:

Given :-

• Roots are 1 and $-\dfrac{1}{2}$

To Find :-

• What is the qradratic equation.

Formula Used :-

$\boxed{\bold{\small{{x}^{2} - (Sum\: of\: roots)x + Products\: of\: roots\: =\: 0}}}$

Solution :-

Given roots :

• α = 1
• β = $-\dfrac{1}{2}$

Now, first we have to find the sum of roots,

➢ Sum of roots = α + β

⇒ $1 + \bigg(-\dfrac{1}{2}\bigg)$

⇒ $1 - \dfrac{1}{2}$

⇒ $\dfrac{2 - 1}{2}$

➠ $\dfrac{1}{2}$

Again, we have to find the product of roots,

➢ Product of roots = α × β

⇒ $1 \times \bigg(- \dfrac{1}{2}\bigg)$

➠ $- \dfrac{1}{2}$

Now, we have to find the qradratic equation,

By using the qradratic equation formula, we get,

⇒ ${x}^{2} - \bigg(\dfrac{1}{2}\bigg)x + \bigg(- \dfrac{1}{2}\bigg)$ = 0

⇒ $\dfrac{2{x}^{2} - x - 1}{2}$ = 0

➠ 2x² - x - 1

$\therefore$ The qradratic equation whose roots are 1 and $-\dfrac{1}{2}$ is $\boxed{\bold{\tt{\large{2{x}^{2} - x - 1}}}}$