Mister Exam
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- Equation:
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Equation (x-87)-27=36
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Equation x^3+2*x^2-9*x-18=0
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Equation 7-4(3x-1)=5(1-2x)
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Equation 5*x^2=35*x
- Express {x} in terms of y where:
- 8*x-16*y=14
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- 15*x-3*y=19
- Derivative of:
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18
- Sum of series:
-
18
- The double integral of:
- 18
-
Identical expressions
- (x+ three)(x- four)- eighteen = eighteen
- (x plus 3)(x minus 4) minus 18 equally 18
- (x plus three)(x minus four) minus eighteen equally eighteen
- x+3x-4-18=18
-
Similar expressions
- (x+3)(x+4)-18=18
- (x+3)(x-4)+18=18
- (x-3)(x-4)-18=18
(x+3)(x-4)-18=18 equation
The teacher will be very surprised to see your correct solution 😉
The solution
You have entered
[src]
(x + 3)*(x - 4) - 18 = 18
$$\left(x - 4\right) \left(x + 3\right) - 18 = 18$$
Detail solution
Move right part of the equation to
left part with negative sign.
The equation is transformed from
$$\left(x - 4\right) \left(x + 3\right) - 18 = 18$$
to
$$\left(\left(x - 4\right) \left(x + 3\right) - 18\right) - 18 = 0$$
Expand the expression in the equation
$$\left(\left(x - 4\right) \left(x + 3\right) - 18\right) - 18 = 0$$
We get the quadratic equation
$$x^{2} - x - 48 = 0$$
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
$$a = 1$$
$$b = -1$$
$$c = -48$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = \frac{1}{2} + \frac{\sqrt{193}}{2}$$
$$x_{2} = \frac{1}{2} - \frac{\sqrt{193}}{2}$$
left part with negative sign.
The equation is transformed from
$$\left(x - 4\right) \left(x + 3\right) - 18 = 18$$
to
$$\left(\left(x - 4\right) \left(x + 3\right) - 18\right) - 18 = 0$$
Expand the expression in the equation
$$\left(\left(x - 4\right) \left(x + 3\right) - 18\right) - 18 = 0$$
We get the quadratic equation
$$x^{2} - x - 48 = 0$$
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
$$a = 1$$
$$b = -1$$
$$c = -48$$
, then
D = b^2 - 4 * a * c =
(-1)^2 - 4 * (1) * (-48) = 193
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = \frac{1}{2} + \frac{\sqrt{193}}{2}$$
$$x_{2} = \frac{1}{2} - \frac{\sqrt{193}}{2}$$
Rapid solution
[src]
_____
1 \/ 193
x1 = - - -------
2 2
$$x_{1} = \frac{1}{2} - \frac{\sqrt{193}}{2}$$
_____
1 \/ 193
x2 = - + -------
2 2
$$x_{2} = \frac{1}{2} + \frac{\sqrt{193}}{2}$$
x2 = 1/2 + sqrt(193)/2
Sum and product of roots
[src]
sum
_____ _____ 1 \/ 193 1 \/ 193 - - ------- + - + ------- 2 2 2 2
$$\left(\frac{1}{2} - \frac{\sqrt{193}}{2}\right) + \left(\frac{1}{2} + \frac{\sqrt{193}}{2}\right)$$
=
1
$$1$$
product
/ _____\ / _____\ |1 \/ 193 | |1 \/ 193 | |- - -------|*|- + -------| \2 2 / \2 2 /
$$\left(\frac{1}{2} - \frac{\sqrt{193}}{2}\right) \left(\frac{1}{2} + \frac{\sqrt{193}}{2}\right)$$
=
-48
$$-48$$
-48