Question

Choose suitable value of p so that the expression
$( {x}^{2} - px + \frac{121}{25} )$
is complete square. ​

Answer 1

$\large\underline{\sf{Given - }}$

$\sf \: A \: quadratic \: expression \: : {x}^{2} - px + \dfrac{121}{25}$

$\large\underline{\sf{To\:Find - }}$

$\sf \: value \: of \: p \: for \: which \: expression \: is \: perfect \: square$

Basic Concept Used :-

• Method of completing squares

$\large\underline{\sf{Solution-}}$

$\sf \: A \: quadratic \: expression \: : {x}^{2} - px + \dfrac{121}{25}$

Add and subtract the square of half the coefficient of x

It means,

$\sf \: Add \: and \: subtract \: \dfrac{ {p}^{2} }{4} \: we \: get$

$= \: \sf \: {x}^{2} - px + \dfrac{121}{25} + \dfrac{ {p}^{2} }{4} - \dfrac{ {p}^{2} }{4}$

$= \: \sf \: {x}^{2} - px + \dfrac{ {p}^{2} }{4} + \dfrac{121}{25}- \dfrac{ {p}^{2} }{4}$

$= \: \sf \: \bigg( {x}^{2} \times 2 \times \dfrac{p}{2} \times x + \dfrac{ {p}^{2} }{4} \bigg) + \dfrac{121}{25}- \dfrac{ {p}^{2} }{4}$

$= \: \sf \: {\bigg( x - \dfrac{p}{2} \bigg) }^{2} + \dfrac{121}{25}- \dfrac{ {p}^{2} }{4}$

Now, expression must be a perfect square, so

$\bf\implies \: \dfrac{121}{25}- \dfrac{ {p}^{2} }{4} = 0$

$\rm \: \dfrac{121}{25} \: = \: \dfrac{ {p}^{2} }{4}$

$\rm \: {p}^{2} = \dfrac{121 \times 4}{25}$

$\rm :\implies\:p \: = \: \pm \: \dfrac{11 \times 2}{5} = \pm \: \dfrac{22}{5}$

$\bf \: Hence, \: value \: of \: p \: = \: \pm \: \dfrac{22}{5}$

Additional Information :-

Nature of roots of Quadratic equation :-

Let us consider a quadratic equation ax² + bx + c = 0, then nature of roots of quadratic equation depends upon Discriminant (D) of the quadratic equation.

• If Discriminant, D > 0, then roots of the equation are real and unequal.

• If Discriminant, D = 0, then roots of the equation are real and equal.

• If Discriminant, D < 0, then roots of the equation are unreal or complex or imaginary.

Where,

• Discriminant, D = b² - 4ac

Different method by which you can solve quadratic equation

There are various methods by which you can solve a quadratic equation such as:

• Factorization

• Completing the square

• Quadratic formula

• Graphical Method

These are the four general methods by which we can solve a quadratic equation.