Mister Exam
Other calculators
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How to use it?
- Equation:
-
Equation sin2x=-1
-
Equation sqrt(x-1)=3-x
-
Equation sin3x=1
-
Equation cot(x)=1
- Express {x} in terms of y where:
- 12*x+14*y=-15
- 4*x-7*y=4
- -6*x-12*y=9
- -10*x-18*y=6
- Graphing y =:
- sqrt(x-1)
- 3-x
- Integral of d{x}:
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sqrt(x-1)
- Derivative of:
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sqrt(x-1)
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3-x
- Limit of the function:
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3-x
-
Identical expressions
- sqrt(x- one)= three -x
- square root of (x minus 1) equally 3 minus x
- square root of (x minus one) equally three minus x
- √(x-1)=3-x
- sqrtx-1=3-x
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Similar expressions
- -0,001*(130cos(sqrtx)-126)=sin(sqrtx)/sqrtx
- 2sin((4п/3)-x)-sin((4п/3)+x)=0
- sqrt(x+1)=3-x
- sqrt((x-1)/(x+1))*((x-1)/(x+1)-1)*(2/(x+1)+((x-1)/(x+1)-1)/(x-1)+2/(x-1))/(4*(x-1))=0
- ((x+9)/3)-(x/7)=2
- sqrt(x-1)=3+x
- x-sqrt(x)-12=0
sqrt(x-1)=3-x equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$\sqrt{x - 1} = 3 - x$$
$$\sqrt{x - 1} = 3 - x$$
We raise the equation sides to 2-th degree
$$x - 1 = \left(3 - x\right)^{2}$$
$$x - 1 = x^{2} - 6 x + 9$$
Transfer the right side of the equation left part with negative sign
$$- x^{2} + 7 x - 10 = 0$$
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
$$a = -1$$
$$b = 7$$
$$c = -10$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = 2$$
$$x_{2} = 5$$
Because
$$\sqrt{x - 1} = 3 - x$$
and
$$\sqrt{x - 1} \geq 0$$
then
$$3 - x \geq 0$$
or
$$x \leq 3$$
$$-\infty < x$$
The final answer:
$$x_{1} = 2$$
$$\sqrt{x - 1} = 3 - x$$
$$\sqrt{x - 1} = 3 - x$$
We raise the equation sides to 2-th degree
$$x - 1 = \left(3 - x\right)^{2}$$
$$x - 1 = x^{2} - 6 x + 9$$
Transfer the right side of the equation left part with negative sign
$$- x^{2} + 7 x - 10 = 0$$
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
$$a = -1$$
$$b = 7$$
$$c = -10$$
, then
D = b^2 - 4 * a * c =
(7)^2 - 4 * (-1) * (-10) = 9
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = 2$$
$$x_{2} = 5$$
Because
$$\sqrt{x - 1} = 3 - x$$
and
$$\sqrt{x - 1} \geq 0$$
then
$$3 - x \geq 0$$
or
$$x \leq 3$$
$$-\infty < x$$
The final answer:
$$x_{1} = 2$$
The graph