Question

11) What equation results from completing the square and then factoring? x^2 -10x =46

Answer 1

$\sf \purple{completing \: the \: square \: method}$

${x}^{2} - 10x = 46$

$\implies {x}^{2} - 10x + {5}^{2} = 46 + 25$

$\implies {(x - 5)}^{2} = 71$

$\implies x - 5 = \pm \sqrt{ 71}$

$\orange{\therefore x= 5\pm \sqrt{ 71}}$

HOPE IT HELPS!!

Answer 2

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

Given Quadratic equation is

$\rm :\longmapsto\: {x}^{2} - 10x = 46$

On adding both sides, the square of half the coefficient of x, i.e. 25, we get

$\rm :\longmapsto\: {x}^{2} - 10x + 25 = 46 + 25$

$\rm :\longmapsto\: {x}^{2} - (2)(5)x + {5}^{2} = 71$

We know,

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

So, using this identity, we get

$\rm :\longmapsto\: {(x - 5)}^{2} = 71$

$\rm :\longmapsto\:x - 5 = \: \pm \: \sqrt{71}$

$\rm :\longmapsto\:x \: = \: 5\: \pm \: \sqrt{71}$

Hence,

$\purple{\rm :\longmapsto\: {x}^{2} - 10x = 46} \\ \rm \: have \: solution \\ \rm :\longmapsto\:x \: = \: 5\: \pm \: \sqrt{71}$

And in factor form

$\boxed{\tt{ {x}^{2} - 10x - 46 = (x - 5 - \sqrt{71})(x - 5 + \sqrt{71})}}$

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MORE TO KNOW

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