Mister Exam
Other calculators
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How to use it?
- Equation:
- Equation x-y=7
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Equation 1/(x+6)=2
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Equation (x-4)^2-x^2=0
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Equation 6/(x+5)=-5
- Express {x} in terms of y where:
- 7*x-1*y=-7
- 10*x-15*y=16
- -14*x-12*y=5
- 13*x-12*y=19
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Identical expressions
- nine *x^ two +y^ two - twelve *x+ eight *y+ twenty-one = zero
- 9 multiply by x squared plus y squared minus 12 multiply by x plus 8 multiply by y plus 21 equally 0
- nine multiply by x to the power of two plus y to the power of two minus twelve multiply by x plus eight multiply by y plus twenty minus one equally zero
- 9*x2+y2-12*x+8*y+21=0
- 9*x²+y²-12*x+8*y+21=0
- 9*x to the power of 2+y to the power of 2-12*x+8*y+21=0
- 9x^2+y^2-12x+8y+21=0
- 9x2+y2-12x+8y+21=0
- 9*x^2+y^2-12*x+8*y+21=O
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Similar expressions
- 9*x^2+y^2+12*x+8*y+21=0
- 9*x^2+y^2-12*x+8*y-21=0
- 9*x^2-y^2-12*x+8*y+21=0
- 9*x^2+y^2-12*x-8*y+21=0
9*x^2+y^2-12*x+8*y+21=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
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
$$a = 9$$
$$b = -12$$
$$c = y^{2} + 8 y + 21$$
, then
The equation has two roots.
or
$$x_{1} = \frac{\sqrt{- 36 y^{2} - 288 y - 612}}{18} + \frac{2}{3}$$
Simplify
$$x_{2} = \frac{2}{3} - \frac{\sqrt{- 36 y^{2} - 288 y - 612}}{18}$$
Simplify
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
$$a = 9$$
$$b = -12$$
$$c = y^{2} + 8 y + 21$$
, then
D = b^2 - 4 * a * c =
(-12)^2 - 4 * (9) * (21 + y^2 + 8*y) = -612 - 288*y - 36*y^2
The equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = \frac{\sqrt{- 36 y^{2} - 288 y - 612}}{18} + \frac{2}{3}$$
Simplify
$$x_{2} = \frac{2}{3} - \frac{\sqrt{- 36 y^{2} - 288 y - 612}}{18}$$
Simplify
Vieta's Theorem
rewrite the equation
$$9 x^{2} - 12 x + y^{2} + 8 y + 21 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - \frac{4 x}{3} + \frac{y^{2}}{9} + \frac{8 y}{9} + \frac{7}{3} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = - \frac{4}{3}$$
$$q = \frac{c}{a}$$
$$q = \frac{y^{2}}{9} + \frac{8 y}{9} + \frac{7}{3}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{4}{3}$$
$$x_{1} x_{2} = \frac{y^{2}}{9} + \frac{8 y}{9} + \frac{7}{3}$$
$$9 x^{2} - 12 x + y^{2} + 8 y + 21 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - \frac{4 x}{3} + \frac{y^{2}}{9} + \frac{8 y}{9} + \frac{7}{3} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = - \frac{4}{3}$$
$$q = \frac{c}{a}$$
$$q = \frac{y^{2}}{9} + \frac{8 y}{9} + \frac{7}{3}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{4}{3}$$
$$x_{1} x_{2} = \frac{y^{2}}{9} + \frac{8 y}{9} + \frac{7}{3}$$
The graph
Rapid solution
[src]
________________
/ 2
2 \/ -17 - y - 8*y
x1 = - - -------------------
3 3
$$x_{1} = \frac{2}{3} - \frac{\sqrt{- y^{2} - 8 y - 17}}{3}$$
________________
/ 2
2 \/ -17 - y - 8*y
x2 = - + -------------------
3 3
$$x_{2} = \frac{\sqrt{- y^{2} - 8 y - 17}}{3} + \frac{2}{3}$$
Sum and product of roots
[src]
sum
________________ ________________
/ 2 / 2
2 \/ -17 - y - 8*y 2 \/ -17 - y - 8*y
0 + - - ------------------- + - + -------------------
3 3 3 3
$$\left(\left(\frac{2}{3} - \frac{\sqrt{- y^{2} - 8 y - 17}}{3}\right) + 0\right) + \left(\frac{\sqrt{- y^{2} - 8 y - 17}}{3} + \frac{2}{3}\right)$$
=
4/3
$$\frac{4}{3}$$
product
/ ________________\ / ________________\ | / 2 | | / 2 | |2 \/ -17 - y - 8*y | |2 \/ -17 - y - 8*y | 1*|- - -------------------|*|- + -------------------| \3 3 / \3 3 /
$$1 \cdot \left(\frac{2}{3} - \frac{\sqrt{- y^{2} - 8 y - 17}}{3}\right) \left(\frac{\sqrt{- y^{2} - 8 y - 17}}{3} + \frac{2}{3}\right)$$
=
2 7 y 8*y - + -- + --- 3 9 9
$$\frac{y^{2}}{9} + \frac{8 y}{9} + \frac{7}{3}$$
7/3 + y^2/9 + 8*y/9