Question

x^2 -x(4+ ✓2) +4✓2 = 0​

Answer 1

$\tt{x^2 -x(4+ \sqrt{2} ) +4 \sqrt{2} = 0}$

$\tt{{x}^{2}-(4+\sqrt{2})x+4\sqrt{2}=0}$

$\tt{x=\frac{4+\sqrt{2}+\sqrt{2(9-4\sqrt{2})}}{2},\frac{4+\sqrt{2}-\sqrt{2(9-4\sqrt{2})}}{2}}$

$\small \tt{x=\frac{1}{\sqrt{2}}+\frac{4+\sqrt{2(9-4\sqrt{2})}}{2},\frac{1}{\sqrt{2}}+\frac{4-\sqrt{2(9-4\sqrt{2})}}{2}}$

Decimal Form: 4, 1.414214

Answer 2

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

Given quadratic equation is

$\rm :\longmapsto\: {x}^{2} - x(4 + \sqrt{2}) + 4 \sqrt{2} = 0$

can be rewritten as

$\rm :\longmapsto\: {x}^{2} - 4x - \sqrt{2}x + 4 \sqrt{2} = 0$

$\rm :\longmapsto\:x(x - 4) - \sqrt{2}(x - 4) = 0$

$\rm :\longmapsto\:(x - 4)(x - \sqrt{2}) = 0$

$\rm :\longmapsto\:x - 4 = 0 \: \: \: \: or \: \: \: \: \: x - \sqrt{2}= 0$

$\bf\implies \:x = 4 \: \: \: \: or \: \: \: \: x = \sqrt{2}$

Verification

$\purple{\rm :\longmapsto\:When \: x = 4}$

The given equation is

$\purple{\rm :\longmapsto\: {x}^{2} - x(4 + \sqrt{2}) + 4 \sqrt{2} = 0}$

On substituting the value of x, we get

$\purple{\rm :\longmapsto\: {4}^{2} - 4(4 + \sqrt{2}) + 4 \sqrt{2} = 0}$

$\purple{\rm :\longmapsto\: 16 - 16 - 4 \sqrt{2} + 4 \sqrt{2} = 0}$

$\purple{\rm\implies \:0 = 0}$

Hence, Verified

$\red{\rm :\longmapsto\:When \: x = \sqrt{2}}$

The given equation is

$\red{\rm :\longmapsto\: {x}^{2} - x(4 + \sqrt{2}) + 4 \sqrt{2} = 0}$

On substituting the value of x, we get

$\red{\rm :\longmapsto\: {( \sqrt{2} )}^{2} - \sqrt{2} (4 + \sqrt{2}) + 4 \sqrt{2} = 0}$

$\red{\rm :\longmapsto\: 2 - 4\sqrt{2} - 2 + 4 \sqrt{2} = 0}$

$\red{\rm\implies \:0 = 0}$

Hence, Verified

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LEARN MORE

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