Question

Find the roots by completing the square 2x2 -7x+3=0​

Answer 1

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

Given quadratic equation is

$\red{\rm :\longmapsto\: {2x}^{2} - 7x + 3 = 0 \: \: }$

can be rewritten as

$\red{\rm :\longmapsto\: {2x}^{2} - 7x = - \: 3 \: \: }$

On dividing each term by 2, we get

$\purple{\rm :\longmapsto\: {x}^{2} - \dfrac{7}{2}x = - \: \dfrac{3}{2} \: \: }$

can be rewritten as

$\purple{\rm :\longmapsto\: {x}^{2} -2 \times \dfrac{7}{4}x = - \: \dfrac{3}{2} \: \: }$

Adding the square of half the coefficient of x, i.e (7/4)^2 on both sides, we get

$\purple{\rm :\longmapsto\: {x}^{2} -2 \times \dfrac{7}{4}x + {\bigg[\dfrac{7}{4} \bigg]}^{2} = - \: \dfrac{3}{2} + {\bigg[\dfrac{7}{4} \bigg]}^{2} \: \: }$

We know,

$\red{\boxed{ \rm{ \: {x}^{2} + {y}^{2} - 2xy ={(x - y)}^{2} \: \: }}}$

So, using this identity, we get

$\purple{\rm :\longmapsto\: {\bigg[x - \dfrac{7}{4} \bigg]}^{2} = - \: \dfrac{3}{2} + \dfrac{49}{16} \: \: }$

$\purple{\rm :\longmapsto\: {\bigg[x - \dfrac{7}{4} \bigg]}^{2} = \dfrac{ - 24 + 49}{16} \: \: }$

$\purple{\rm :\longmapsto\: {\bigg[x - \dfrac{7}{4} \bigg]}^{2} = \dfrac{25}{16} \: \: }$

$\green{\rm :\longmapsto\:x - \dfrac{7}{4} = \: \pm \: \dfrac{5}{4} \: \: }$

$\green{\rm :\longmapsto\:x = \dfrac{7}{4} \: \: \pm \: \dfrac{5}{4} \: \: }$

$\green{\rm :\longmapsto\:x = \dfrac{7 \: \pm \: 5}{4}}$

$\green{\rm :\longmapsto\:x = \dfrac{7 \: + \: 5}{4} \: \: \: or \: \: \: \dfrac{7 \: - \: 5}{4} }$

$\green{\rm :\longmapsto\:x = \dfrac{12}{4} \: \: \: or \: \: \: \dfrac{2}{4} }$

$\bf\implies \:x = 3 \: \: \: or \: \: \: x \: = \: \dfrac{1}{2}$

Additional Information:-

Nature of roots :-

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