Mister Exam
Other calculators
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How to use it?
- Equation:
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Equation (x^2+3)/(x^2+1)=2
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Equation -4x+9=-19
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Equation -4x+3=2x
- Equation 3*x^3-27*x=0
- Express {x} in terms of y where:
- 13*x-13*y=-4
- 19*x-20*y=-12
- 12*x+7*y=15
- -2*x-9*y=3
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Identical expressions
- (x- three)(nine -x)= one
- (x minus 3)(9 minus x) equally 1
- (x minus three)(nine minus x) equally one
- x-39-x=1
-
Similar expressions
- (x+3)(9-x)=1
- (x-3)(9+x)=1
(x-3)(9-x)=1 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Move right part of the equation to
left part with negative sign.
The equation is transformed from
$$\left(9 - x\right) \left(x - 3\right) = 1$$
to
$$\left(9 - x\right) \left(x - 3\right) - 1 = 0$$
Expand the expression in the equation
$$\left(9 - x\right) \left(x - 3\right) - 1 = 0$$
We get the quadratic equation
$$- x^{2} + 12 x - 28 = 0$$
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
$$a = -1$$
$$b = 12$$
$$c = -28$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = 6 - 2 \sqrt{2}$$
$$x_{2} = 2 \sqrt{2} + 6$$
left part with negative sign.
The equation is transformed from
$$\left(9 - x\right) \left(x - 3\right) = 1$$
to
$$\left(9 - x\right) \left(x - 3\right) - 1 = 0$$
Expand the expression in the equation
$$\left(9 - x\right) \left(x - 3\right) - 1 = 0$$
We get the quadratic equation
$$- x^{2} + 12 x - 28 = 0$$
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
$$a = -1$$
$$b = 12$$
$$c = -28$$
, then
D = b^2 - 4 * a * c =
(12)^2 - 4 * (-1) * (-28) = 32
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = 6 - 2 \sqrt{2}$$
$$x_{2} = 2 \sqrt{2} + 6$$
Rapid solution
[src]
___ x1 = 6 - 2*\/ 2
$$x_{1} = 6 - 2 \sqrt{2}$$
___ x2 = 6 + 2*\/ 2
$$x_{2} = 2 \sqrt{2} + 6$$
x2 = 2*sqrt(2) + 6
Sum and product of roots
[src]
sum
___ ___ 6 - 2*\/ 2 + 6 + 2*\/ 2
$$\left(6 - 2 \sqrt{2}\right) + \left(2 \sqrt{2} + 6\right)$$
=
12
$$12$$
product
/ ___\ / ___\ \6 - 2*\/ 2 /*\6 + 2*\/ 2 /
$$\left(6 - 2 \sqrt{2}\right) \left(2 \sqrt{2} + 6\right)$$
=
28
$$28$$
28