Mister Exam
Other calculators
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How to use it?
- Equation:
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Equation x^2+9=(x+9)^2
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Equation cos(x)=-1
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Equation 3*x+50=77
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Equation 5-2(x-1)=4-x
- Express {x} in terms of y where:
- 10*x+5*y=-17
- 14*x+16*y=-3
- -19*x+1*y=0
- 6*x-2*y=-10
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Identical expressions
- - two *x+ four *x+ five = two *x^ two - seven *x+ two *x- three
- minus 2 multiply by x plus 4 multiply by x plus 5 equally 2 multiply by x squared minus 7 multiply by x plus 2 multiply by x minus 3
- minus two multiply by x plus four multiply by x plus five equally two multiply by x to the power of two minus seven multiply by x plus two multiply by x minus three
- -2*x+4*x+5=2*x2-7*x+2*x-3
- -2*x+4*x+5=2*x²-7*x+2*x-3
- -2*x+4*x+5=2*x to the power of 2-7*x+2*x-3
- -2x+4x+5=2x^2-7x+2x-3
- -2x+4x+5=2x2-7x+2x-3
-
Similar expressions
- -2*x+4*x-5=2*x^2-7*x+2*x-3
- -2*x+4*x+5=2*x^2-7*x-2*x-3
- -2*x-4*x+5=2*x^2-7*x+2*x-3
- 2*x+4*x+5=2*x^2-7*x+2*x-3
- -2*x+4*x+5=2*x^2-7*x+2*x+3
- -2*x+4*x+5=2*x^2+7*x+2*x-3
-2*x+4*x+5=2*x^2-7*x+2*x-3 equation
The teacher will be very surprised to see your correct solution 😉
The solution
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[src]
2 -2*x + 4*x + 5 = 2*x - 7*x + 2*x - 3
$$- 2 x + 4 x + 5 = 2 x^{2} - 7 x + 2 x - 3$$
Detail solution
Move right part of the equation to
left part with negative sign.
The equation is transformed from
$$- 2 x + 4 x + 5 = 2 x^{2} - 7 x + 2 x - 3$$
to
$$\left(- 2 x + 4 x + 5\right) + \left(- 2 x^{2} - 2 x + 7 x + 3\right) = 0$$
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
$$a = -2$$
$$b = 7$$
$$c = 8$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = \frac{7}{4} - \frac{\sqrt{113}}{4}$$
Simplify
$$x_{2} = \frac{7}{4} + \frac{\sqrt{113}}{4}$$
Simplify
left part with negative sign.
The equation is transformed from
$$- 2 x + 4 x + 5 = 2 x^{2} - 7 x + 2 x - 3$$
to
$$\left(- 2 x + 4 x + 5\right) + \left(- 2 x^{2} - 2 x + 7 x + 3\right) = 0$$
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
$$a = -2$$
$$b = 7$$
$$c = 8$$
, then
D = b^2 - 4 * a * c =
(7)^2 - 4 * (-2) * (8) = 113
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = \frac{7}{4} - \frac{\sqrt{113}}{4}$$
Simplify
$$x_{2} = \frac{7}{4} + \frac{\sqrt{113}}{4}$$
Simplify
Vieta's Theorem
rewrite the equation
$$- 2 x + 4 x + 5 = 2 x^{2} - 7 x + 2 x - 3$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - \frac{7 x}{2} - 4 = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = - \frac{7}{2}$$
$$q = \frac{c}{a}$$
$$q = -4$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{7}{2}$$
$$x_{1} x_{2} = -4$$
$$- 2 x + 4 x + 5 = 2 x^{2} - 7 x + 2 x - 3$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - \frac{7 x}{2} - 4 = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = - \frac{7}{2}$$
$$q = \frac{c}{a}$$
$$q = -4$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{7}{2}$$
$$x_{1} x_{2} = -4$$
Rapid solution
[src]
_____
7 \/ 113
x1 = - - -------
4 4
$$x_{1} = \frac{7}{4} - \frac{\sqrt{113}}{4}$$
_____
7 \/ 113
x2 = - + -------
4 4
$$x_{2} = \frac{7}{4} + \frac{\sqrt{113}}{4}$$
Sum and product of roots
[src]
sum
_____ _____
7 \/ 113 7 \/ 113
0 + - - ------- + - + -------
4 4 4 4
$$\left(\left(\frac{7}{4} - \frac{\sqrt{113}}{4}\right) + 0\right) + \left(\frac{7}{4} + \frac{\sqrt{113}}{4}\right)$$
=
7/2
$$\frac{7}{2}$$
product
/ _____\ / _____\ |7 \/ 113 | |7 \/ 113 | 1*|- - -------|*|- + -------| \4 4 / \4 4 /
$$1 \cdot \left(\frac{7}{4} - \frac{\sqrt{113}}{4}\right) \left(\frac{7}{4} + \frac{\sqrt{113}}{4}\right)$$
=
-4
$$-4$$
-4
The graph