Answer

Verified

63.6k+ views

**Hint:**In this question, we are given that the roots of the quadratic equation \[{x^2} + px + q = 0\;\] differ by$1$. We have to calculate the sum and the product of the roots, i.e., $\alpha + \beta = \dfrac{{ - B}}{A}$, $\alpha \beta = \dfrac{C}{A}$ where the equation is \[A{x^2} + Bx + C = 0\]. Then apply the algebraic identity ${\left( {\alpha + \beta } \right)^2} = {\left( {\alpha - \beta } \right)^2} + 4\alpha \beta$ and put in the required values.

**Formula Used:**

Quadratic equations in general: \[A{x^2} + Bx + C = 0\]

Let, the roots of the above quadratic equation be $\alpha$ and $\beta$

Therefore, the sum of roots is $\alpha + \beta = \dfrac{{ - B}}{A}$ and the product of roots is $\alpha \beta = \dfrac{C}{A}$

The algebraic identity is${\left( {\alpha + \beta } \right)^2} = {\left( {\alpha - \beta } \right)^2} + 4\alpha \beta$

**Complete step by step Solution:**

Given that, \[{x^2} + px + q = 0\;\], is a quadratic equation.

Let, the roots of the given equation be $\alpha$ and $\beta$

According to the question, the roots of the equation differ by$1$.

It implies that $\alpha - \beta = 1$ ------- (1)

Now, compare the equation \[{x^2} + px + q = 0\;\]with the general quadratic equation, i.e., \[A{x^2} + Bx + C = 0\].

We obtain$A = 1$,$B = p$, and$C = q$.

Using the formula for sum and the product of the roots, the sum of the roots,

\[\alpha + \beta = \dfrac{{ - B}}{A}\]

It implies that,

\[\alpha + \beta = - p\] --------(2)

And the product of the roots, $\alpha \beta = \dfrac{C}{A}$

$\Rightarrow \alpha \beta = q$ --------(3)

Now, applying the algebraic identity ${\left( {\alpha + \beta } \right)^2} = {\left( {\alpha - \beta } \right)^2} + 4\alpha \beta$

Substituting the required values in the above equation, (from equations (1), (2), and (3)),

It will be ${\left( { - p} \right)^2} = {\left( 1 \right)^2} + 4q$

On solving, we get ${p^2} = 4q+ 1$

**Hence, the correct option is (B).**

**Note:**The values of $x$ that fulfil a given quadratic equation \[A{x^2} + Bx + C = 0\] are known as its roots. They are, in other words, the values of the variables $\left( x \right)$ that satisfy the equation. The roots of a quadratic function are the $x -$coordinates of the function's $x -$intercepts. Because the degree of a quadratic equation is $2$, it can only have two roots.

Recently Updated Pages

Write a composition in approximately 450 500 words class 10 english JEE_Main

Arrange the sentences P Q R between S1 and S5 such class 10 english JEE_Main

What is the common property of the oxides CONO and class 10 chemistry JEE_Main

What happens when dilute hydrochloric acid is added class 10 chemistry JEE_Main

If four points A63B 35C4 2 and Dx3x are given in such class 10 maths JEE_Main

The area of square inscribed in a circle of diameter class 10 maths JEE_Main