1. Find one pair $(x,y)$ of real numbers such that $x + y = 4$ and $x^3 + y^3 = 100.$
2.For what real values of $k$ does the quadratic $12x^2 + kx + 27 = 0$ have nonreal roots? Enter your answer as an interval.
3.Find all pairs $(x,y)$ of real numbers such that $x + y = 10$ and $x^2 + y^2 = 56$.
4.Simplify $\displaystyle\frac{1-i}{2+3i}$, where $i^2 = -1.$
5.What is the smallest value of $x$ that satisfies the equation $8x^2 - 38x + 35 = 0$? Express your answer as a decimal.

Question

13-01-2022
Mathematics

contestada

1. Find one pair $(x,y)$ of real numbers such that $x + y = 4$ and $x^3 + y^3 = 100.$
2.For what real values of $k$ does the quadratic $12x^2 + kx + 27 = 0$ have nonreal roots? Enter your answer as an interval.
3.Find all pairs $(x,y)$ of real numbers such that $x + y = 10$ and $x^2 + y^2 = 56$.
4.Simplify $\displaystyle\frac{1-i}{2+3i}$, where $i^2 = -1.$
5.What is the smallest value of $x$ that satisfies the equation $8x^2 - 38x + 35 = 0$? Express your answer as a decimal.

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xero099 xero099 · Answer 1 · 2022-01-19T19:36:47+01:00

1) The pairs [tex](x, y) = (2 + \sqrt{7}, 2 - \sqrt{7})[/tex] and [tex](x, y) = (2 - \sqrt{7}, 2 + \sqrt{7})[/tex] are solutions of the system.

2) The quadratic formula has conjugated complex roots for [tex]k \in (-36, 36)[/tex].

3) The pairs [tex](x, y) = (5 + \sqrt{47}, 5 - \sqrt{47})[/tex] and [tex](x, y) = (5 - \sqrt{47}, 5 + \sqrt{47})[/tex] are solutions of the system.

4) The complex number [tex]z = \frac{1-i}{2+3\cdot i}[/tex] is equal to the complex number [tex]-\frac{1}{13}-\frac{5}{13}\cdot i[/tex].

5) [tex]1.25[/tex] is the smallest value that satisfies the quadratic equation [tex]8\cdot x^{2}-38\cdot x + 35 = 0[/tex].

Procedure - Miscellaneous on quadratic functions and complex numbers

1) Pair of real numbers within a system of equations (I)

We need to solve the following system of equations to determine at least one pair of real numbers that are its solution:

[tex]x+y = 4[/tex] (1)

[tex]x^{3} + y^{3} = 100[/tex] (2)

By (1) in (2) we have the following expression.

[tex](4-y)^{3} + y^{3} = 100[/tex]

[tex]y^{2}-48\cdot y -36 = 0[/tex] (3)

Whose solutions are: [tex]y_{1} = 2 + \sqrt{7}[/tex], [tex]y_{2} = 2 - \sqrt{7}[/tex]

And by (1) we find the respective solutions for [tex]x[/tex]: [tex]x_{1} = 2-\sqrt{7}[/tex], [tex]x_{2} = 2 +\sqrt{7}[/tex].

In a nutshell, the pairs [tex](x, y) = (2 + \sqrt{7}, 2 - \sqrt{7})[/tex] and [tex](x, y) = (2 - \sqrt{7}, 2 + \sqrt{7})[/tex] are solutions of the system. [tex]\blacksquare[/tex]

2) Real values associated to conjugated complex roots of a quadratic formula

By the Quadratic Formula we understand that roots of

[tex]12\cdot x^{2}+k\cdot x + 27 = 0[/tex] are conjugated complex if and only if the following condition is observed:

[tex]d^{2} = k^{2}-1296 < 0[/tex] (4)

Where [tex]d[/tex] is the discriminant of the quadratic formula.

After some mathematical handling we have the following result:

[tex]k^{2} < 1296[/tex]

[tex]-36 < k < 36[/tex]

Hence, the quadratic formula has conjugated complex roots for [tex]k \in (-36, 36)[/tex]. [tex]\blacksquare[/tex]

3) Pair of real numbers within a system of equations (II)

By using the approach used in part 1), we find that the resulting polynomial is [tex]2\cdot y^{2} -20\cdot y -44 = 0[/tex] for [tex]x = 10-y[/tex], whose solutions are [tex](x,y) = (5 + \sqrt{47}, 5 - \sqrt{47})[/tex] and [tex](x,y) = (5-\sqrt{47}, 5+\sqrt{47})[/tex].

In a nutshell, the pairs [tex](x, y) = (5 + \sqrt{47}, 5 - \sqrt{47})[/tex] and [tex](x, y) = (5 - \sqrt{47}, 5 + \sqrt{47})[/tex] are solutions of the system. [tex]\blacksquare[/tex]

4) Simplification of a complex number

Let be [tex]z = \frac{1-i}{2+3\cdot i}[/tex], we proceed to simplify the expression by means of complex algebra:

[tex]\frac{1-i}{2+3\cdot i} = \frac{(1-i)\cdot (2-3\cdot i)}{(2+3\cdot i)\cdot (2-3\cdot i)} = \frac{2-5\cdot i + 3\cdot i^{2}}{2^{2}+3^{2}} = -\frac{1}{13} -\frac{5}{13} \cdot i[/tex]

The complex number [tex]z = \frac{1-i}{2+3\cdot i}[/tex] is equal to the complex number [tex]-\frac{1}{13}-\frac{5}{13}\cdot i[/tex]. [tex]\blacksquare[/tex]

5) Determination of the least root by the quadratic formula

Let be [tex]8\cdot x^{2}-38\cdot x + 35 = 0[/tex], whose roots are contained in the following quadratic formula:

[tex]x = \frac{38\pm \sqrt{(-38)^{2}-4\cdot (8)\cdot (35)}}{2\cdot (8)}[/tex]

Whose solutions are: [tex]x_{1} = 3.5[/tex] and [tex]x_{2} = 1.25[/tex]. We notice that the latter root is the smallest value of [tex]x[/tex]. In consequence, we conclude that [tex]1.25[/tex] is the smallest value that satisfies the quadratic equation [tex]8\cdot x^{2}-38\cdot x + 35 = 0[/tex]. [tex]\blacksquare[/tex]

Remark

The statement is poorly formatted. Correct form is presented below:

Find one pair [tex](x,y)[/tex] of real numbers such that [tex]x+y = 4[/tex] and [tex]x^{3} + y^{3} = 100[/tex].
For what real values of [tex]k[/tex] does the quadratic [tex]12\cdot x^{2}+k\cdot x + 27 = 0[/tex] have nonreal roots? Enter your answer as an interval.
Find all pairs [tex](x,y)[/tex] of real numbers such that [tex]x+y = 10[/tex] and [tex]x^{2}+y^{2} = 56[/tex].
Simplify [tex]\frac{1-i}{2+3\cdot i}[/tex] where [tex]i^{2} = -1[/tex].
What is the smallest value of [tex]x[/tex] that satisfies the equation [tex]8\cdot x^{2}-38\cdot x + 35 = 0[/tex]? Express your answer as a decimal.

To learn more on quadratic functions, we kindly invite to check this verified question: https://brainly.com/question/4119784

To learn more on complex numbers, we kindly invite to check this verified question: https://brainly.com/question/10251853