Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation x^2-4x+6=0
-
Equation 5*x=12
-
Equation x-1=sqrt(x+5)
-
Equation x-4/2+3*x=5
- Express {x} in terms of y where:
- -20*x+2*y=-12
- 4*x-2*y=4
- -3*x+11*y=5
- 6*x-8*y=11
- Graphing y =:
- x-1
- Canonical form:
- x-1
- Derivative of:
-
x-1
- Integral of d{x}:
- sqrt(x+5)
-
Identical expressions
- x- one =sqrt(x+ five)
- x minus 1 equally square root of (x plus 5)
- x minus one equally square root of (x plus five)
- x-1=√(x+5)
- x-1=sqrtx+5
-
Similar expressions
- sqrt((x+5)/x)+4*sqrt(x/(x+5))=4
- sqrtx+57-sqrt(20+x)=1
- sqrt(x+5*sqrt(x-25))+sqrt(x-5*sqrt(x+25))=10
- x+1=sqrt(x+5)
- x-1=sqrt(x-5)
x-1=sqrt(x+5) equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$x - 1 = \sqrt{x + 5}$$
Transfer the right side of the equation left part with negative sign
$$- \sqrt{x + 5} = 1 - x$$
We raise the equation sides to 2-th degree
$$x + 5 = \left(1 - x\right)^{2}$$
$$x + 5 = x^{2} - 2 x + 1$$
Transfer the right side of the equation left part with negative sign
$$- x^{2} + 3 x + 4 = 0$$
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
$$a = -1$$
$$b = 3$$
$$c = 4$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = -1$$
$$x_{2} = 4$$
Because
$$\sqrt{x + 5} = x - 1$$
and
$$\sqrt{x + 5} \geq 0$$
then
$$x - 1 \geq 0$$
or
$$1 \leq x$$
$$x < \infty$$
The final answer:
$$x_{2} = 4$$
$$x - 1 = \sqrt{x + 5}$$
Transfer the right side of the equation left part with negative sign
$$- \sqrt{x + 5} = 1 - x$$
We raise the equation sides to 2-th degree
$$x + 5 = \left(1 - x\right)^{2}$$
$$x + 5 = x^{2} - 2 x + 1$$
Transfer the right side of the equation left part with negative sign
$$- x^{2} + 3 x + 4 = 0$$
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
$$a = -1$$
$$b = 3$$
$$c = 4$$
, then
D = b^2 - 4 * a * c =
(3)^2 - 4 * (-1) * (4) = 25
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = -1$$
$$x_{2} = 4$$
Because
$$\sqrt{x + 5} = x - 1$$
and
$$\sqrt{x + 5} \geq 0$$
then
$$x - 1 \geq 0$$
or
$$1 \leq x$$
$$x < \infty$$
The final answer:
$$x_{2} = 4$$
The graph