Question

if a quadratic equation x²‐px+q has equal roots the p/2=0​

Answer 1

Proper Question

If a quadratic equation $x^2-px+q=0$ has equal roots, then $\dfrac{p}{2}$ is the two solutions.

Before We Solve

Here, we will use the formula of Vieta. Vieta's formula can obtain the sum and the product of roots, using the coefficients of the equation. So, let's do it!

Solution

Given equation: $x^2-px+q=0$

Firstly, let's talk about what is Vieta's formula. Vieta's formula is used to find the sum and product of all roots in an equation. Let's take a cubic equation as an example.

$(x-\alpha )(x-\beta )(x-\gamma )=ax^3+bx^2+cx+d$

$\rightarrow x^3-(\alpha +\beta +\gamma )x^2+(\alpha \beta +\beta \gamma +\gamma \alpha )x-\alpha \beta \gamma=ax^3+bx^2+cx+d$

By comparing the coefficients, we can see that,

Sum of roots

$=\alpha +\beta +\gamma=-\dfrac{b}{a}$

Product of roots

$=\alpha \beta \gamma =-\dfrac{d}{a}$

Now, we will apply this notion to our answer.

Firstly, assuming the two roots as $x=\alpha ,\beta$ we get,

$\begin{cases} & \alpha+\beta=-\dfrac{-p}{1} \\ & \alpha\beta=\dfrac{q}{1} \end{cases}$

So,

$\begin{cases} & \alpha+\beta=p \\ & \alpha\beta=q \end{cases}$

Here, the two roots are equal. So, we can choose the arithmetic mean as two equal solutions.

$\therefore\alpha ,\beta =\dfrac{p}{2}$

Conclusion

So, we get the roots as $x=\dfrac{p}{2}\ \textsf{(Double Roots)}$.