Mister Exam
Other calculators
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How to use it?
- Equation:
-
Equation 5x^2+9x+4=0
-
Equation (x-4)/(x-6)=2
-
Equation e^x=2
-
Equation 4*x+5=2*x-7
- Express {x} in terms of y where:
- -15*x+9*y=-6
- -17*x+11*y=-1
- -11*x+4*y=-20
- 3*x-1*y=5
- Derivative of:
- 4x^2+7x
-
Identical expressions
- 4x^ two + seven x=7+24x
- 4x squared plus 7x equally 7 plus 24x
- 4x to the power of two plus seven x equally 7 plus 24x
- 4x2+7x=7+24x
- 4x²+7x=7+24x
- 4x to the power of 2+7x=7+24x
-
Similar expressions
- 4x^2-7x=7+24x
- 4x^2+7x=7-24x
- 32*x^9-48*x^7+24*x^5-8*x^3=0
- 4*x2+7=7+24*x
4x^2+7x=7+24x equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Move right part of the equation to
left part with negative sign.
The equation is transformed from
$$4 x^{2} + 7 x = 24 x + 7$$
to
$$\left(- 24 x - 7\right) + \left(4 x^{2} + 7 x\right) = 0$$
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
$$a = 4$$
$$b = -17$$
$$c = -7$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = \frac{17}{8} + \frac{\sqrt{401}}{8}$$
$$x_{2} = \frac{17}{8} - \frac{\sqrt{401}}{8}$$
left part with negative sign.
The equation is transformed from
$$4 x^{2} + 7 x = 24 x + 7$$
to
$$\left(- 24 x - 7\right) + \left(4 x^{2} + 7 x\right) = 0$$
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
$$a = 4$$
$$b = -17$$
$$c = -7$$
, then
D = b^2 - 4 * a * c =
(-17)^2 - 4 * (4) * (-7) = 401
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = \frac{17}{8} + \frac{\sqrt{401}}{8}$$
$$x_{2} = \frac{17}{8} - \frac{\sqrt{401}}{8}$$
Vieta's Theorem
rewrite the equation
$$4 x^{2} + 7 x = 24 x + 7$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - \frac{17 x}{4} - \frac{7}{4} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = - \frac{17}{4}$$
$$q = \frac{c}{a}$$
$$q = - \frac{7}{4}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{17}{4}$$
$$x_{1} x_{2} = - \frac{7}{4}$$
$$4 x^{2} + 7 x = 24 x + 7$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - \frac{17 x}{4} - \frac{7}{4} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = - \frac{17}{4}$$
$$q = \frac{c}{a}$$
$$q = - \frac{7}{4}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{17}{4}$$
$$x_{1} x_{2} = - \frac{7}{4}$$
Rapid solution
[src]
_____
17 \/ 401
x1 = -- - -------
8 8
$$x_{1} = \frac{17}{8} - \frac{\sqrt{401}}{8}$$
_____
17 \/ 401
x2 = -- + -------
8 8
$$x_{2} = \frac{17}{8} + \frac{\sqrt{401}}{8}$$
x2 = 17/8 + sqrt(401)/8
Sum and product of roots
[src]
sum
_____ _____ 17 \/ 401 17 \/ 401 -- - ------- + -- + ------- 8 8 8 8
$$\left(\frac{17}{8} - \frac{\sqrt{401}}{8}\right) + \left(\frac{17}{8} + \frac{\sqrt{401}}{8}\right)$$
=
17/4
$$\frac{17}{4}$$
product
/ _____\ / _____\ |17 \/ 401 | |17 \/ 401 | |-- - -------|*|-- + -------| \8 8 / \8 8 /
$$\left(\frac{17}{8} - \frac{\sqrt{401}}{8}\right) \left(\frac{17}{8} + \frac{\sqrt{401}}{8}\right)$$
=
-7/4
$$- \frac{7}{4}$$
-7/4