Mister Exam
Other calculators
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How to use it?
- Equation:
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Equation (x+2)^2=(1-x)^2
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Equation 2*cos(x)+1=0
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Equation (-5*x-3)*(2*x-1)=0
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Equation x^2-8x+15=0
- Express {x} in terms of y where:
- 7*x+13*y=-13
- 7*x+19*y=12
- 1*x+20*y=-17
- 14*x-16*y=-3
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Identical expressions
- (three)/(8y^ three + one)-(one)/(two y+ one)=(y+ three)/(y^2-2y+ one)
- (3) divide by (8y cubed plus 1) minus (1) divide by (2y plus 1) equally (y plus 3) divide by (y squared minus 2y plus 1)
- (three) divide by (8y to the power of three plus one) minus (one) divide by (two y plus one) equally (y plus three) divide by (y squared minus 2y plus one)
- (3)/(8y3+1)-(1)/(2y+1)=(y+3)/(y2-2y+1)
- 3/8y3+1-1/2y+1=y+3/y2-2y+1
- (3)/(8y³+1)-(1)/(2y+1)=(y+3)/(y²-2y+1)
- (3)/(8y to the power of 3+1)-(1)/(2y+1)=(y+3)/(y to the power of 2-2y+1)
- 3/8y^3+1-1/2y+1=y+3/y^2-2y+1
- (3) divide by (8y^3+1)-(1) divide by (2y+1)=(y+3) divide by (y^2-2y+1)
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Similar expressions
- (3)/(8y^3+1)-(1)/(2y+1)=(y+3)/(y^2+2y+1)
- (3)/(8y^3-1)-(1)/(2y+1)=(y+3)/(y^2-2y+1)
- (3)/(8y^3+1)+(1)/(2y+1)=(y+3)/(y^2-2y+1)
- (3)/(8y^3+1)-(1)/(2y-1)=(y+3)/(y^2-2y+1)
- (3)/(8y^3+1)-(1)/(2y+1)=(y-3)/(y^2-2y+1)
- (3)/(8y^3+1)-(1)/(2y+1)=(y+3)/(y^2-2y-1)
(3)/(8y^3+1)-(1)/(2y+1)=(y+3)/(y^2-2y+1) equation
The teacher will be very surprised to see your correct solution 😉
The solution
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3 1 y + 3 -------- - ------- = ------------ 3 2*y + 1 2 8*y + 1 y - 2*y + 1
$$\frac{3}{8 y^{3} + 1} - \frac{1}{2 y + 1} = \frac{y + 3}{\left(y^{2} - 2 y\right) + 1}$$
Sum and product of roots
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sum
______________ / ______________ \ ______________ / ______________ \ ______________
/ ____ | / ____ | / ____ | / ____ | / ____
/ 221 \/ 67 | ___ / 221 \/ 67 | / 221 \/ 67 | ___ / 221 \/ 67 | / 221 \/ 67
3 / --- + ------ |\/ 3 *3 / --- + ------ ___ | 3 / --- + ------ | \/ 3 *3 / --- + ------ ___ | 3 / --- + ------
2 1 \/ 108 4 | \/ 108 4 \/ 3 | 2 1 \/ 108 4 | \/ 108 4 \/ 3 | 2 \/ 108 4 1
- - - ----------------------- + ------------------- + I*|------------------------- + -----------------------| + - - - ----------------------- + ------------------- + I*|- ------------------------- - -----------------------| + - - - ------------------- + ----------------------
9 ______________ 6 | 6 ______________| 9 ______________ 6 | 6 ______________| 9 3 ______________
/ ____ | / ____ | / ____ | / ____ | / ____
/ 221 \/ 67 | / 221 \/ 67 | / 221 \/ 67 | / 221 \/ 67 | / 221 \/ 67
108*3 / --- + ------ | 108*3 / --- + ------ | 108*3 / --- + ------ | 108*3 / --- + ------ | 54*3 / --- + ------
\/ 108 4 \ \/ 108 4 / \/ 108 4 \ \/ 108 4 / \/ 108 4
$$\left(- \frac{\sqrt[3]{\frac{221}{108} + \frac{\sqrt{67}}{4}}}{3} - \frac{2}{9} + \frac{1}{54 \sqrt[3]{\frac{221}{108} + \frac{\sqrt{67}}{4}}}\right) + \left(\left(- \frac{2}{9} - \frac{1}{108 \sqrt[3]{\frac{221}{108} + \frac{\sqrt{67}}{4}}} + \frac{\sqrt[3]{\frac{221}{108} + \frac{\sqrt{67}}{4}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{221}{108} + \frac{\sqrt{67}}{4}}}{6} - \frac{\sqrt{3}}{108 \sqrt[3]{\frac{221}{108} + \frac{\sqrt{67}}{4}}}\right)\right) + \left(- \frac{2}{9} - \frac{1}{108 \sqrt[3]{\frac{221}{108} + \frac{\sqrt{67}}{4}}} + \frac{\sqrt[3]{\frac{221}{108} + \frac{\sqrt{67}}{4}}}{6} + i \left(\frac{\sqrt{3}}{108 \sqrt[3]{\frac{221}{108} + \frac{\sqrt{67}}{4}}} + \frac{\sqrt{3} \sqrt[3]{\frac{221}{108} + \frac{\sqrt{67}}{4}}}{6}\right)\right)\right)$$
=
/ ______________ \ / ______________ \
| / ____ | | / ____ |
| ___ / 221 \/ 67 | | ___ / 221 \/ 67 |
| \/ 3 *3 / --- + ------ ___ | |\/ 3 *3 / --- + ------ ___ |
2 | \/ 108 4 \/ 3 | | \/ 108 4 \/ 3 |
- - + I*|- ------------------------- - -----------------------| + I*|------------------------- + -----------------------|
3 | 6 ______________| | 6 ______________|
| / ____ | | / ____ |
| / 221 \/ 67 | | / 221 \/ 67 |
| 108*3 / --- + ------ | | 108*3 / --- + ------ |
\ \/ 108 4 / \ \/ 108 4 /
$$- \frac{2}{3} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{221}{108} + \frac{\sqrt{67}}{4}}}{6} - \frac{\sqrt{3}}{108 \sqrt[3]{\frac{221}{108} + \frac{\sqrt{67}}{4}}}\right) + i \left(\frac{\sqrt{3}}{108 \sqrt[3]{\frac{221}{108} + \frac{\sqrt{67}}{4}}} + \frac{\sqrt{3} \sqrt[3]{\frac{221}{108} + \frac{\sqrt{67}}{4}}}{6}\right)$$
product
/ ______________ / ______________ \\ / ______________ / ______________ \\ / ______________ \ | / ____ | / ____ || | / ____ | / ____ || | / ____ | | / 221 \/ 67 | ___ / 221 \/ 67 || | / 221 \/ 67 | ___ / 221 \/ 67 || | / 221 \/ 67 | | 3 / --- + ------ |\/ 3 *3 / --- + ------ ___ || | 3 / --- + ------ | \/ 3 *3 / --- + ------ ___ || | 3 / --- + ------ | | 2 1 \/ 108 4 | \/ 108 4 \/ 3 || | 2 1 \/ 108 4 | \/ 108 4 \/ 3 || | 2 \/ 108 4 1 | |- - - ----------------------- + ------------------- + I*|------------------------- + -----------------------||*|- - - ----------------------- + ------------------- + I*|- ------------------------- - -----------------------||*|- - - ------------------- + ----------------------| | 9 ______________ 6 | 6 ______________|| | 9 ______________ 6 | 6 ______________|| | 9 3 ______________| | / ____ | / ____ || | / ____ | / ____ || | / ____ | | / 221 \/ 67 | / 221 \/ 67 || | / 221 \/ 67 | / 221 \/ 67 || | / 221 \/ 67 | | 108*3 / --- + ------ | 108*3 / --- + ------ || | 108*3 / --- + ------ | 108*3 / --- + ------ || | 54*3 / --- + ------ | \ \/ 108 4 \ \/ 108 4 // \ \/ 108 4 \ \/ 108 4 // \ \/ 108 4 /
$$\left(- \frac{2}{9} - \frac{1}{108 \sqrt[3]{\frac{221}{108} + \frac{\sqrt{67}}{4}}} + \frac{\sqrt[3]{\frac{221}{108} + \frac{\sqrt{67}}{4}}}{6} + i \left(\frac{\sqrt{3}}{108 \sqrt[3]{\frac{221}{108} + \frac{\sqrt{67}}{4}}} + \frac{\sqrt{3} \sqrt[3]{\frac{221}{108} + \frac{\sqrt{67}}{4}}}{6}\right)\right) \left(- \frac{2}{9} - \frac{1}{108 \sqrt[3]{\frac{221}{108} + \frac{\sqrt{67}}{4}}} + \frac{\sqrt[3]{\frac{221}{108} + \frac{\sqrt{67}}{4}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{221}{108} + \frac{\sqrt{67}}{4}}}{6} - \frac{\sqrt{3}}{108 \sqrt[3]{\frac{221}{108} + \frac{\sqrt{67}}{4}}}\right)\right) \left(- \frac{\sqrt[3]{\frac{221}{108} + \frac{\sqrt{67}}{4}}}{3} - \frac{2}{9} + \frac{1}{54 \sqrt[3]{\frac{221}{108} + \frac{\sqrt{67}}{4}}}\right)$$
=
-1/6
$$- \frac{1}{6}$$
-1/6
Rapid solution
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______________ / ______________ \
/ ____ | / ____ |
/ 221 \/ 67 | ___ / 221 \/ 67 |
3 / --- + ------ |\/ 3 *3 / --- + ------ ___ |
2 1 \/ 108 4 | \/ 108 4 \/ 3 |
y1 = - - - ----------------------- + ------------------- + I*|------------------------- + -----------------------|
9 ______________ 6 | 6 ______________|
/ ____ | / ____ |
/ 221 \/ 67 | / 221 \/ 67 |
108*3 / --- + ------ | 108*3 / --- + ------ |
\/ 108 4 \ \/ 108 4 /
$$y_{1} = - \frac{2}{9} - \frac{1}{108 \sqrt[3]{\frac{221}{108} + \frac{\sqrt{67}}{4}}} + \frac{\sqrt[3]{\frac{221}{108} + \frac{\sqrt{67}}{4}}}{6} + i \left(\frac{\sqrt{3}}{108 \sqrt[3]{\frac{221}{108} + \frac{\sqrt{67}}{4}}} + \frac{\sqrt{3} \sqrt[3]{\frac{221}{108} + \frac{\sqrt{67}}{4}}}{6}\right)$$
______________ / ______________ \
/ ____ | / ____ |
/ 221 \/ 67 | ___ / 221 \/ 67 |
3 / --- + ------ | \/ 3 *3 / --- + ------ ___ |
2 1 \/ 108 4 | \/ 108 4 \/ 3 |
y2 = - - - ----------------------- + ------------------- + I*|- ------------------------- - -----------------------|
9 ______________ 6 | 6 ______________|
/ ____ | / ____ |
/ 221 \/ 67 | / 221 \/ 67 |
108*3 / --- + ------ | 108*3 / --- + ------ |
\/ 108 4 \ \/ 108 4 /
$$y_{2} = - \frac{2}{9} - \frac{1}{108 \sqrt[3]{\frac{221}{108} + \frac{\sqrt{67}}{4}}} + \frac{\sqrt[3]{\frac{221}{108} + \frac{\sqrt{67}}{4}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{221}{108} + \frac{\sqrt{67}}{4}}}{6} - \frac{\sqrt{3}}{108 \sqrt[3]{\frac{221}{108} + \frac{\sqrt{67}}{4}}}\right)$$
______________
/ ____
/ 221 \/ 67
3 / --- + ------
2 \/ 108 4 1
y3 = - - - ------------------- + ----------------------
9 3 ______________
/ ____
/ 221 \/ 67
54*3 / --- + ------
\/ 108 4
$$y_{3} = - \frac{\sqrt[3]{\frac{221}{108} + \frac{\sqrt{67}}{4}}}{3} - \frac{2}{9} + \frac{1}{54 \sqrt[3]{\frac{221}{108} + \frac{\sqrt{67}}{4}}}$$
y3 = -(221/108 + sqrt(67)/4)^(1/3)/3 - 2/9 + 1/(54*(221/108 + sqrt(67)/4)^(1/3))
Numerical answer
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y1 = 0.0385827652803931 + 0.471779867138517*i
y2 = 0.0385827652803931 - 0.471779867138517*i
y3 = -0.743832197227453
y3 = -0.743832197227453