Mister Exam
Other calculators
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How to use it?
- Equation:
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Equation sin(pi*x)=0
-
Equation 14x-25=20x+9
-
Equation (2x-1)^2-4x^2=0
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Equation (5x-8)(4x-3)=0
- Express {x} in terms of y where:
- 9*x-8*y=6
- 15*x+15*y=16
- -4*x+18*y=-14
- 8*x+7*y=-13
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Identical expressions
- x^ three -9x^ two +26x+ twenty-four = zero
- x cubed minus 9x squared plus 26x plus 24 equally 0
- x to the power of three minus 9x to the power of two plus 26x plus twenty minus four equally zero
- x3-9x2+26x+24=0
- x³-9x²+26x+24=0
- x to the power of 3-9x to the power of 2+26x+24=0
- x^3-9x^2+26x+24=O
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Similar expressions
- x^3-9x^2+26x-24=0
- x^3+9x^2+26x+24=0
- x^3-9x^2-26x+24=0
x^3-9x^2+26x+24=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Vieta's Theorem
it is reduced cubic equation
$$p x^{2} + x^{3} + q x + v = 0$$
where
$$p = \frac{b}{a}$$
$$p = -9$$
$$q = \frac{c}{a}$$
$$q = 26$$
$$v = \frac{d}{a}$$
$$v = 24$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 9$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 26$$
$$x_{1} x_{2} x_{3} = 24$$
$$p x^{2} + x^{3} + q x + v = 0$$
where
$$p = \frac{b}{a}$$
$$p = -9$$
$$q = \frac{c}{a}$$
$$q = 26$$
$$v = \frac{d}{a}$$
$$v = 24$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 9$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 26$$
$$x_{1} x_{2} x_{3} = 24$$
Sum and product of roots
[src]
sum
/ 2/3\ / 2/3\
| / _______\ | | / _______\ |
_________________ 6 ___ | 1 \648 + 3*\/ 46653 / | _________________ 6 ___ |1 \648 + 3*\/ 46653 / | _________________
3 ___ 3 / _______ 2/3 I*\/ 3 *|- - + ----------------------| 3 ___ 3 / _______ 2/3 I*\/ 3 *|- - ----------------------| 3 ___ 3 / _______ 2/3
\/ 3 *\/ 216 + \/ 46653 3 \ 2 6 / \/ 3 *\/ 216 + \/ 46653 3 \2 6 / \/ 3 *\/ 216 + \/ 46653 3
3 + -------------------------- + ---------------------- + -------------------------------------- + 3 + -------------------------- + ---------------------- + ------------------------------------ + 3 - -------------------------- - ----------------------
6 _________________ _________________ 6 _________________ _________________ 3 _________________
3 / _______ 3 / _______ 3 / _______ 3 / _______ 3 / _______
6*\/ 216 + \/ 46653 \/ 216 + \/ 46653 6*\/ 216 + \/ 46653 \/ 216 + \/ 46653 3*\/ 216 + \/ 46653
$$\left(\frac{3^{\frac{2}{3}}}{6 \sqrt[3]{\sqrt{46653} + 216}} + \frac{\sqrt[3]{3} \sqrt[3]{\sqrt{46653} + 216}}{6} + 3 + \frac{\sqrt[6]{3} i \left(- \frac{1}{2} + \frac{\left(3 \sqrt{46653} + 648\right)^{\frac{2}{3}}}{6}\right)}{\sqrt[3]{\sqrt{46653} + 216}}\right) + \left(\frac{3^{\frac{2}{3}}}{6 \sqrt[3]{\sqrt{46653} + 216}} + \frac{\sqrt[3]{3} \sqrt[3]{\sqrt{46653} + 216}}{6} + 3 + \frac{\sqrt[6]{3} i \left(- \frac{\left(3 \sqrt{46653} + 648\right)^{\frac{2}{3}}}{6} + \frac{1}{2}\right)}{\sqrt[3]{\sqrt{46653} + 216}}\right) + \left(- \frac{\sqrt[3]{3} \sqrt[3]{\sqrt{46653} + 216}}{3} - \frac{3^{\frac{2}{3}}}{3 \sqrt[3]{\sqrt{46653} + 216}} + 3\right)$$
=
/ 2/3\ / 2/3\
| / _______\ | | / _______\ |
6 ___ |1 \648 + 3*\/ 46653 / | 6 ___ | 1 \648 + 3*\/ 46653 / |
I*\/ 3 *|- - ----------------------| I*\/ 3 *|- - + ----------------------|
\2 6 / \ 2 6 /
9 + ------------------------------------ + --------------------------------------
_________________ _________________
3 / _______ 3 / _______
\/ 216 + \/ 46653 \/ 216 + \/ 46653
$$9 + \frac{\sqrt[6]{3} i \left(- \frac{\left(3 \sqrt{46653} + 648\right)^{\frac{2}{3}}}{6} + \frac{1}{2}\right)}{\sqrt[3]{\sqrt{46653} + 216}} + \frac{\sqrt[6]{3} i \left(- \frac{1}{2} + \frac{\left(3 \sqrt{46653} + 648\right)^{\frac{2}{3}}}{6}\right)}{\sqrt[3]{\sqrt{46653} + 216}}$$
product
/ 2/3\ / 2/3\
| / _______\ | | / _______\ |
_________________ 6 ___ | 1 \648 + 3*\/ 46653 / | _________________ 6 ___ |1 \648 + 3*\/ 46653 / | _________________
3 ___ 3 / _______ 2/3 I*\/ 3 *|- - + ----------------------| 3 ___ 3 / _______ 2/3 I*\/ 3 *|- - ----------------------| 3 ___ 3 / _______ 2/3
\/ 3 *\/ 216 + \/ 46653 3 \ 2 6 / \/ 3 *\/ 216 + \/ 46653 3 \2 6 / \/ 3 *\/ 216 + \/ 46653 3
3 + -------------------------- + ---------------------- + -------------------------------------- * 3 + -------------------------- + ---------------------- + ------------------------------------ * 3 - -------------------------- - ----------------------
6 _________________ _________________ 6 _________________ _________________ 3 _________________
3 / _______ 3 / _______ 3 / _______ 3 / _______ 3 / _______
6*\/ 216 + \/ 46653 \/ 216 + \/ 46653 6*\/ 216 + \/ 46653 \/ 216 + \/ 46653 3*\/ 216 + \/ 46653
$$\left(\frac{3^{\frac{2}{3}}}{6 \sqrt[3]{\sqrt{46653} + 216}} + \frac{\sqrt[3]{3} \sqrt[3]{\sqrt{46653} + 216}}{6} + 3 + \frac{\sqrt[6]{3} i \left(- \frac{1}{2} + \frac{\left(3 \sqrt{46653} + 648\right)^{\frac{2}{3}}}{6}\right)}{\sqrt[3]{\sqrt{46653} + 216}}\right) * \left(\frac{3^{\frac{2}{3}}}{6 \sqrt[3]{\sqrt{46653} + 216}} + \frac{\sqrt[3]{3} \sqrt[3]{\sqrt{46653} + 216}}{6} + 3 + \frac{\sqrt[6]{3} i \left(- \frac{\left(3 \sqrt{46653} + 648\right)^{\frac{2}{3}}}{6} + \frac{1}{2}\right)}{\sqrt[3]{\sqrt{46653} + 216}}\right) * \left(- \frac{\sqrt[3]{3} \sqrt[3]{\sqrt{46653} + 216}}{3} - \frac{3^{\frac{2}{3}}}{3 \sqrt[3]{\sqrt{46653} + 216}} + 3\right)$$
=
-24
$$-24$$
Rapid solution
[src]
/ 2/3\
| / _______\ |
_________________ 6 ___ | 1 \648 + 3*\/ 46653 / |
3 ___ 3 / _______ 2/3 I*\/ 3 *|- - + ----------------------|
\/ 3 *\/ 216 + \/ 46653 3 \ 2 6 /
x_1 = 3 + -------------------------- + ---------------------- + --------------------------------------
6 _________________ _________________
3 / _______ 3 / _______
6*\/ 216 + \/ 46653 \/ 216 + \/ 46653
$$x_{1} = \frac{3^{\frac{2}{3}}}{6 \sqrt[3]{\sqrt{46653} + 216}} + \frac{\sqrt[3]{3} \sqrt[3]{\sqrt{46653} + 216}}{6} + 3 + \frac{\sqrt[6]{3} i \left(- \frac{1}{2} + \frac{\left(3 \sqrt{46653} + 648\right)^{\frac{2}{3}}}{6}\right)}{\sqrt[3]{\sqrt{46653} + 216}}$$
/ 2/3\
| / _______\ |
_________________ 6 ___ |1 \648 + 3*\/ 46653 / |
3 ___ 3 / _______ 2/3 I*\/ 3 *|- - ----------------------|
\/ 3 *\/ 216 + \/ 46653 3 \2 6 /
x_2 = 3 + -------------------------- + ---------------------- + ------------------------------------
6 _________________ _________________
3 / _______ 3 / _______
6*\/ 216 + \/ 46653 \/ 216 + \/ 46653
$$x_{2} = \frac{3^{\frac{2}{3}}}{6 \sqrt[3]{\sqrt{46653} + 216}} + \frac{\sqrt[3]{3} \sqrt[3]{\sqrt{46653} + 216}}{6} + 3 + \frac{\sqrt[6]{3} i \left(- \frac{\left(3 \sqrt{46653} + 648\right)^{\frac{2}{3}}}{6} + \frac{1}{2}\right)}{\sqrt[3]{\sqrt{46653} + 216}}$$
_________________
3 ___ 3 / _______ 2/3
\/ 3 *\/ 216 + \/ 46653 3
x_3 = 3 - -------------------------- - ----------------------
3 _________________
3 / _______
3*\/ 216 + \/ 46653
$$x_{3} = - \frac{\sqrt[3]{3} \sqrt[3]{\sqrt{46653} + 216}}{3} - \frac{3^{\frac{2}{3}}}{3 \sqrt[3]{\sqrt{46653} + 216}} + 3$$
Numerical answer
[src]
x1 = 4.86297120224424 - 3.06789587521708*i
x2 = -0.72594240448847
x3 = 4.86297120224424 + 3.06789587521708*i
x3 = 4.86297120224424 + 3.06789587521708*i
The graph