Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation x^2=11
-
Equation 2*x^2+6*x+9=0
-
Equation 10(x-9)=7
-
Equation (x+2)^4+(x+2)^2-12=0
- Express {x} in terms of y where:
- -6*x-17*y=14
- 4*x-18*y=-14
- -14*x-1*y=3
- 1*x-6*y=-19
- Integral of d{x}:
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x-7
- Graphing y =:
- x-7
- Derivative of:
-
x-7
-
Identical expressions
- sqrt(x- six)=x- seven
- square root of (x minus 6) equally x minus 7
- square root of (x minus six) equally x minus seven
- √(x-6)=x-7
- sqrtx-6=x-7
-
Similar expressions
- sqrtx-6-x^3-1=0
- sqrt(x-6)=x+7
- x-(sqrtx)-6=0
- sqrt(x)-6=sqrt(4)-x
- sqrt(x+6)=x-7
sqrt(x-6)=x-7 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$\sqrt{x - 6} = x - 7$$
$$\sqrt{x - 6} = x - 7$$
We raise the equation sides to 2-th degree
$$x - 6 = \left(x - 7\right)^{2}$$
$$x - 6 = x^{2} - 14 x + 49$$
Transfer the right side of the equation left part with negative sign
$$- x^{2} + 15 x - 55 = 0$$
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
$$a = -1$$
$$b = 15$$
$$c = -55$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = \frac{15}{2} - \frac{\sqrt{5}}{2}$$
$$x_{2} = \frac{\sqrt{5}}{2} + \frac{15}{2}$$
Because
$$\sqrt{x - 6} = x - 7$$
and
$$\sqrt{x - 6} \geq 0$$
then
$$x - 7 \geq 0$$
or
$$7 \leq x$$
$$x < \infty$$
The final answer:
$$x_{2} = \frac{\sqrt{5}}{2} + \frac{15}{2}$$
$$\sqrt{x - 6} = x - 7$$
$$\sqrt{x - 6} = x - 7$$
We raise the equation sides to 2-th degree
$$x - 6 = \left(x - 7\right)^{2}$$
$$x - 6 = x^{2} - 14 x + 49$$
Transfer the right side of the equation left part with negative sign
$$- x^{2} + 15 x - 55 = 0$$
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
$$a = -1$$
$$b = 15$$
$$c = -55$$
, then
D = b^2 - 4 * a * c =
(15)^2 - 4 * (-1) * (-55) = 5
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = \frac{15}{2} - \frac{\sqrt{5}}{2}$$
$$x_{2} = \frac{\sqrt{5}}{2} + \frac{15}{2}$$
Because
$$\sqrt{x - 6} = x - 7$$
and
$$\sqrt{x - 6} \geq 0$$
then
$$x - 7 \geq 0$$
or
$$7 \leq x$$
$$x < \infty$$
The final answer:
$$x_{2} = \frac{\sqrt{5}}{2} + \frac{15}{2}$$
Sum and product of roots
[src]
sum
___ 15 \/ 5 -- + ----- 2 2
$$\frac{\sqrt{5}}{2} + \frac{15}{2}$$
=
___ 15 \/ 5 -- + ----- 2 2
$$\frac{\sqrt{5}}{2} + \frac{15}{2}$$
product
___ 15 \/ 5 -- + ----- 2 2
$$\frac{\sqrt{5}}{2} + \frac{15}{2}$$
=
___ 15 \/ 5 -- + ----- 2 2
$$\frac{\sqrt{5}}{2} + \frac{15}{2}$$
15/2 + sqrt(5)/2
Rapid solution
[src]
___
15 \/ 5
x1 = -- + -----
2 2
$$x_{1} = \frac{\sqrt{5}}{2} + \frac{15}{2}$$
x1 = sqrt(5)/2 + 15/2
The graph