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Identical expressions
- sqrt(x)- four root(4)(x)= five
- square root of (x) minus 4root(4)(x) equally 5
- square root of (x) minus four root(4)(x) equally five
- √(x)-4root(4)(x)=5
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Similar expressions
- sqrt(x)+4root(4)(x)=5
sqrt(x)-4root(4)(x)=5 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$\sqrt{x} - 4 \sqrt{4} x = 5$$
$$\sqrt{x} = 8 x + 5$$
We raise the equation sides to 2-th degree
$$x = \left(8 x + 5\right)^{2}$$
$$x = 64 x^{2} + 80 x + 25$$
Transfer the right side of the equation left part with negative sign
$$- 64 x^{2} - 79 x - 25 = 0$$
This equation is of the form
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = -64$$
$$b = -79$$
$$c = -25$$
, then
Because D<0, then the equation
has no real roots,
but complex roots is exists.
or
$$x_{1} = - \frac{79}{128} - \frac{\sqrt{159} i}{128}$$
$$x_{2} = - \frac{79}{128} + \frac{\sqrt{159} i}{128}$$
$$\sqrt{x} - 4 \sqrt{4} x = 5$$
$$\sqrt{x} = 8 x + 5$$
We raise the equation sides to 2-th degree
$$x = \left(8 x + 5\right)^{2}$$
$$x = 64 x^{2} + 80 x + 25$$
Transfer the right side of the equation left part with negative sign
$$- 64 x^{2} - 79 x - 25 = 0$$
This equation is of the form
a*x^2 + b*x + c = 0
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = -64$$
$$b = -79$$
$$c = -25$$
, then
D = b^2 - 4 * a * c =
(-79)^2 - 4 * (-64) * (-25) = -159
Because D<0, then the equation
has no real roots,
but complex roots is exists.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = - \frac{79}{128} - \frac{\sqrt{159} i}{128}$$
$$x_{2} = - \frac{79}{128} + \frac{\sqrt{159} i}{128}$$
Rapid solution
[src]
_____
79 I*\/ 159
x1 = - --- - ---------
128 128
$$x_{1} = - \frac{79}{128} - \frac{\sqrt{159} i}{128}$$
_____
79 I*\/ 159
x2 = - --- + ---------
128 128
$$x_{2} = - \frac{79}{128} + \frac{\sqrt{159} i}{128}$$
x2 = -79/128 + sqrt(159)*i/128
Sum and product of roots
[src]
sum
_____ _____ 79 I*\/ 159 79 I*\/ 159 - --- - --------- + - --- + --------- 128 128 128 128
$$\left(- \frac{79}{128} - \frac{\sqrt{159} i}{128}\right) + \left(- \frac{79}{128} + \frac{\sqrt{159} i}{128}\right)$$
=
-79 ---- 64
$$- \frac{79}{64}$$
product
/ _____\ / _____\ | 79 I*\/ 159 | | 79 I*\/ 159 | |- --- - ---------|*|- --- + ---------| \ 128 128 / \ 128 128 /
$$\left(- \frac{79}{128} - \frac{\sqrt{159} i}{128}\right) \left(- \frac{79}{128} + \frac{\sqrt{159} i}{128}\right)$$
=
25 -- 64
$$\frac{25}{64}$$
25/64
Numerical answer
[src]
x1 = -0.6171875 - 0.0985118766634257*i
x2 = -0.6171875 + 0.0985118766634257*i
x2 = -0.6171875 + 0.0985118766634257*i