Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation sin(x)=4
-
Equation 5^x=125
-
Equation (|x+4|)=5
-
Equation 2*sin(x)=1
- Express {x} in terms of y where:
- 16*x-17*y=-15
- -4*x-4*y=19
- 15*x+1*y=19
- -5*x-2*y=9
- Derivative of:
- sqrt(3*x+1)
-
x-1
- Graphing y =:
- sqrt(3*x+1)
- x-1
- Canonical form:
- x-1
-
Identical expressions
- sqrt(three *x+ one)=x- one
- square root of (3 multiply by x plus 1) equally x minus 1
- square root of (three multiply by x plus one) equally x minus one
- √(3*x+1)=x-1
- sqrt(3x+1)=x-1
- sqrt3x+1=x-1
-
Similar expressions
- sqrt(3x+1)=9-x
- sqrt3-x=sqrt3x+1
- 3x-(10-9x)=22x
- sqrt(3*x-1)=x-1
- sqrt(3*x+15)=0
- sqrt(3*x+1)=x+1
- sqrt3x+1=-9
sqrt(3*x+1)=x-1 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$\sqrt{3 x + 1} = x - 1$$
$$\sqrt{3 x + 1} = x - 1$$
We raise the equation sides to 2-th degree
$$3 x + 1 = \left(x - 1\right)^{2}$$
$$3 x + 1 = x^{2} - 2 x + 1$$
Transfer the right side of the equation left part with negative sign
$$- x^{2} + 5 x = 0$$
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
$$a = -1$$
$$b = 5$$
$$c = 0$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = 0$$
$$x_{2} = 5$$
Because
$$\sqrt{3 x + 1} = x - 1$$
and
$$\sqrt{3 x + 1} \geq 0$$
then
$$x - 1 \geq 0$$
or
$$1 \leq x$$
$$x < \infty$$
The final answer:
$$x_{2} = 5$$
$$\sqrt{3 x + 1} = x - 1$$
$$\sqrt{3 x + 1} = x - 1$$
We raise the equation sides to 2-th degree
$$3 x + 1 = \left(x - 1\right)^{2}$$
$$3 x + 1 = x^{2} - 2 x + 1$$
Transfer the right side of the equation left part with negative sign
$$- x^{2} + 5 x = 0$$
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
$$a = -1$$
$$b = 5$$
$$c = 0$$
, then
D = b^2 - 4 * a * c =
(5)^2 - 4 * (-1) * (0) = 25
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = 0$$
$$x_{2} = 5$$
Because
$$\sqrt{3 x + 1} = x - 1$$
and
$$\sqrt{3 x + 1} \geq 0$$
then
$$x - 1 \geq 0$$
or
$$1 \leq x$$
$$x < \infty$$
The final answer:
$$x_{2} = 5$$