Question

solve the following equation for x, giving correct to 3 decimal places:

1) x^2-16x+6=0

Answer 1

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

Given quadratic equation is

$\rm :\longmapsto\: {x}^{2} - 16x + 6 = 0$

On comparing with ax² + bx + c = 0, we get

$\red{\rm :\longmapsto\:a = 1}$

$\red{\rm :\longmapsto\:b = - 16}$

$\red{\rm :\longmapsto\:c = 6}$

We know, the roots of the quadratic equation are evaluated by using Quadratic formula which are as follow

$\red{\rm :\longmapsto\:\boxed{\tt{ x = \frac{ - b \: \pm \: \sqrt{ {b}^{2} - 4ac} }{2a}}}}$

So, on substituting the values of a, b and c, we get

$\rm :\longmapsto\:x = \dfrac{ - ( - 16) \: \pm \: \sqrt{ {( - 16)}^{2} - 4(1)(6) } }{2}$

$\rm :\longmapsto\:x = \dfrac{16 \: \pm \: \sqrt{ 256 - 24 } }{2}$

$\rm :\longmapsto\:x = \dfrac{16 \: \pm \: \sqrt{2 \times 2 \times 58} }{2}$

$\rm :\longmapsto\:x = \dfrac{16 \: \pm \: 2\sqrt{58} }{2}$

$\rm :\longmapsto\:x = \dfrac{2(8\: \pm \: \sqrt{58}) }{2}$

$\rm :\longmapsto\:x = 8 \: \pm \: \sqrt{58}$

$\rm :\longmapsto\:x = 8 \: \pm \: 7.615$

$\rm\implies \:x = 8 + 7.615 \: \: or \: \: 8 - 7.615$

$\rm\implies \:x = 15.615 \: \: or \: \: 0.385$

Note :-

$\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{}}}&{\underline{\sf{\:\:7.615 \:\:}}}\\ {\underline{\sf{7}}}& {\sf{\:\:58.000000 \:\:}} \\{\sf{}}& \underline{\sf{\:\: \: \: \: \: \: \: \: 49 \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \: \:\:}} \\ {\underline{\sf{146}}}& {\sf{\:\: \: \: \: \: 900 \:  \:  \:  \:   \:  \:  \:  \:\:}} \\{\sf{}}& \underline{\sf{\:\: 876 \:  \:  \:  \:  \:  \: \:\:}} \\ {\underline{\sf{1521}}}& {\sf{\:\:2400 \: \: \: \:   \:\:}} \\{\sf{}}& \underline{\sf{\:\:1521 \: \: \: \: \:\:}} \\ {\underline{\sf{15225}}}& {\sf{\:\:  87900\:\:}} \\{\sf{}}& \underline{\sf{\:\: \: 76125\:\:}} \\ {\underline{\sf{}}}& {\sf{\: \: 11775\:\:}}{\sf{}}&{\sf{\:\:\:\:}}\end{array}\end{gathered}$

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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