Question

If the values ​​p = 1 and q = –2 are the roots of the quadratic equation then the quadratic equation is *

1️⃣ x² + 2x –1 = 0
2️⃣ x² - x - 2 = 0
3️⃣ x² - 2x + 1 = 0
4️⃣ x² + x + 2 = 0​

Answer 1

$\Large{\bf{\green{\mathfrak{\dag{\underline{\underline{Given:-}}}}}}}$

• Zeros of a polynomial are 1 and - 2

$\Large{\bf{\orange{\mathfrak{\dag{\underline{\underline{To \: Find:-}}}}}}}$

• The Quadratic equation

$\Large{\bf{\red{\mathfrak{\dag{\underline{\underline{Solution:-}}}}}}}$

We know that,

• Any Quadratic equation given with roots $\alpha$ and $\beta$ is of the form,

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

• Here,

With given zeros p and q,

$\implies{\blue{\sf x^2 \: - \: (p \: + \: q) \: x \: + \: pq \: = \: 0}}$

• Given that,

• p = 1

• q = - 2

Substituting the values,

• x² - (1 + (- 2)) x + 1 * - 2 = 0

• x² - (1 - 2) x - 2 = 0

• x² - (- 1) x - 2 = 0

• x² + 1x - 2 = 0

• x² + x - 2 = 0

Therefore,

• $\underline{\boxed{\purple{\tt{Quadratic \: equation \: = \: x^2 \: + \: x \: - \: 2 \: = \: 0}}}}$

$\Large{\bf{\blue{\mathfrak{\dag{\underline{\underline{Extra \: Information:-}}}}}}}$

• Any quadratic polynomial is of the form ax² + bx + c = 0.

• Any Quadratic equation with given zeros p and q is of the form x² - (p + q) x + pq.

Answer 2

Given :-

• The values of p = 1 and q = - 2 are the roots of the quadratic equation.

To Find :-

• What is the quadratic equation.

Formula Used :-

$\clubsuit$ General Formula For Quadratic Equation :

$\longmapsto \sf\boxed{\bold{\pink{{x}^{2} - (\alpha + \beta)x + \alpha\beta =\: 0}}}$

Solution :-

Given :

• p = 1
• q = - 2

Hence, we can write as :

$\implies \sf {x}^{2} - (p + q)x + pq =\: 0$

By putting the value of p = 1 and q = - 2 we get,

$\implies \sf {x}^{2} - (1 + \{ - 2\})x + 1 \times (- 2) =\: 0$

$\implies \sf {x}^{2} - (1 - 2)x + ( - 2) =\: 0$

$\implies \sf {x}^{2} - 1x + 2x - 2 =\: 0$

$\implies \sf {x}^{2} + 1x - 2 =\: 0$

Now, we can write as,

$\implies\sf\bold{\red{{x}^{2} + x - 2 =\: 0}}$

$\therefore$ The quadratic equation is x² + x - 2 = 0.