Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation (x+6)^3=25*(x+6)
-
Equation 1-5x=-6x+8
-
Equation 1-5*x=-6*x+8
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Equation (x-2)^2=(x-9)^2
- Express {x} in terms of y where:
- -7*x-7*y=-14
- 10*x-2*y=0
- -15*x+17*y=13
- -11*x-15*y=14
- Derivative of:
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x^2+3x-4
- Graphing y =:
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x^2+3x-4
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Identical expressions
- x^ two +3x- four =(seven + two hundred and seventy-three ^ zero . five)/ two
- x squared plus 3x minus 4 equally (7 plus 273 to the power of 0.5) divide by 2
- x to the power of two plus 3x minus four equally (seven plus two hundred and seventy minus three to the power of zero . five) divide by two
- x2+3x-4=(7+2730.5)/2
- x2+3x-4=7+2730.5/2
- x²+3x-4=(7+273^0.5)/2
- x to the power of 2+3x-4=(7+273 to the power of 0.5)/2
- x^2+3x-4=7+273^0.5/2
- x^2+3x-4=(7+273^0.5) divide by 2
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Similar expressions
- x^2+3x+4=(7+273^0.5)/2
- x^2-3x-4=(7+273^0.5)/2
- x^2+3x-4=(7-273^0.5)/2
x^2+3x-4=(7+273^0.5)/2 equation
The teacher will be very surprised to see your correct solution 😉
The solution
You have entered
[src]
_____
2 7 + \/ 273
x + 3*x - 4 = -----------
2
$$\left(x^{2} + 3 x\right) - 4 = \frac{7 + \sqrt{273}}{2}$$
Detail solution
Move right part of the equation to
left part with negative sign.
The equation is transformed from
$$\left(x^{2} + 3 x\right) - 4 = \frac{7 + \sqrt{273}}{2}$$
to
$$\left(\left(x^{2} + 3 x\right) - 4\right) - \frac{7 + \sqrt{273}}{2} = 0$$
Expand the expression in the equation
$$\left(\left(x^{2} + 3 x\right) - 4\right) - \frac{7 + \sqrt{273}}{2} = 0$$
We get the quadratic equation
$$x^{2} + 3 x - \frac{\sqrt{273}}{2} - \frac{15}{2} = 0$$
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
$$a = 1$$
$$b = 3$$
$$c = - \frac{\sqrt{273}}{2} - \frac{15}{2}$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = - \frac{3}{2} + \frac{\sqrt{2 \sqrt{273} + 39}}{2}$$
$$x_{2} = - \frac{\sqrt{2 \sqrt{273} + 39}}{2} - \frac{3}{2}$$
left part with negative sign.
The equation is transformed from
$$\left(x^{2} + 3 x\right) - 4 = \frac{7 + \sqrt{273}}{2}$$
to
$$\left(\left(x^{2} + 3 x\right) - 4\right) - \frac{7 + \sqrt{273}}{2} = 0$$
Expand the expression in the equation
$$\left(\left(x^{2} + 3 x\right) - 4\right) - \frac{7 + \sqrt{273}}{2} = 0$$
We get the quadratic equation
$$x^{2} + 3 x - \frac{\sqrt{273}}{2} - \frac{15}{2} = 0$$
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
$$a = 1$$
$$b = 3$$
$$c = - \frac{\sqrt{273}}{2} - \frac{15}{2}$$
, then
D = b^2 - 4 * a * c =
(3)^2 - 4 * (1) * (-15/2 - sqrt(273)/2) = 39 + 2*sqrt(273)
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = - \frac{3}{2} + \frac{\sqrt{2 \sqrt{273} + 39}}{2}$$
$$x_{2} = - \frac{\sqrt{2 \sqrt{273} + 39}}{2} - \frac{3}{2}$$
Vieta's Theorem
it is reduced quadratic equation
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 3$$
$$q = \frac{c}{a}$$
$$q = - \frac{7 + \sqrt{273}}{2} - 4$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = -3$$
$$x_{1} x_{2} = - \frac{7 + \sqrt{273}}{2} - 4$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 3$$
$$q = \frac{c}{a}$$
$$q = - \frac{7 + \sqrt{273}}{2} - 4$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = -3$$
$$x_{1} x_{2} = - \frac{7 + \sqrt{273}}{2} - 4$$
Sum and product of roots
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sum
________________ ________________
/ _____ / _____
3 \/ 39 + 2*\/ 273 3 \/ 39 + 2*\/ 273
- - + ------------------- + - - - -------------------
2 2 2 2
$$\left(- \frac{\sqrt{2 \sqrt{273} + 39}}{2} - \frac{3}{2}\right) + \left(- \frac{3}{2} + \frac{\sqrt{2 \sqrt{273} + 39}}{2}\right)$$
=
-3
$$-3$$
product
/ ________________\ / ________________\ | / _____ | | / _____ | | 3 \/ 39 + 2*\/ 273 | | 3 \/ 39 + 2*\/ 273 | |- - + -------------------|*|- - - -------------------| \ 2 2 / \ 2 2 /
$$\left(- \frac{3}{2} + \frac{\sqrt{2 \sqrt{273} + 39}}{2}\right) \left(- \frac{\sqrt{2 \sqrt{273} + 39}}{2} - \frac{3}{2}\right)$$
=
_____ 15 \/ 273 - -- - ------- 2 2
$$- \frac{\sqrt{273}}{2} - \frac{15}{2}$$
-15/2 - sqrt(273)/2
Rapid solution
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________________
/ _____
3 \/ 39 + 2*\/ 273
x1 = - - + -------------------
2 2
$$x_{1} = - \frac{3}{2} + \frac{\sqrt{2 \sqrt{273} + 39}}{2}$$
________________
/ _____
3 \/ 39 + 2*\/ 273
x2 = - - - -------------------
2 2
$$x_{2} = - \frac{\sqrt{2 \sqrt{273} + 39}}{2} - \frac{3}{2}$$
x2 = -sqrt(2*sqrt(273) + 39)/2 - 3/2