All questions with tag [math: algebra-precalculus]


Solutions to $z^3 - (b+6) z^2 + 8 b^2 z - 7+b^2 = 0, b\in \mathbb R, z \in \mathbb C$

$z_1 = 1+i$ is a given solution. I guess what I have to find is $z_2$ and $z_3$ in $(z - (1 + i))(z - z_2)(z-z_3) = z^3 - (b+6) z^2 + 8 b^2 z - 7+b^2$. I tried to divide the polynomial by $(z - (1 + i))$, but that didn’t seem to work because of the $b$. According to the Complex conjugate root theorem $z_2 = \overline{z_1} = 1 - i$ is a solution ...
2022-07-25 17:46:44

Analytical method for root finding

Is there a logical method to locate the roots of the list below formula? $$y = -\frac{1}{2}{x}^{2}-\cos(x)+1.1$$ I'm sorry for the unimportant inquiry, I'm new at mathematics!
2022-07-25 17:46:40

How would you solve this system of equations?

I have this system of equations: $$\left\{\begin{align*} &x+y+z+u=5\\ &y+z+u+v = 1\\ &z+u+v+x = 2\\ &u+v+x+y = 0\\ &v+x+y+z = 4 \end{align*}\right.$$ How do I address this system?
2022-07-25 17:46:25

finding ratio of elements when 2 solids are melted to form another solid

Here is my question Suppose if a person has two solids.Solid A which is made up of two elements, element X and element Y and similarly there is also Solid B which is also made up of same two elements.A consists of elements X and Y in ratio 4:9 and B consists of element X and Y in ratio 5:6.Now if equal amount of A and B are melted to form anothe...
2022-07-25 17:46:03

System of equations - what are the values of the coefficients of a quadratic parabola?

In the procedure of relearning the mathematical essentials I'm stumbling over this problem: A square parabola $y = ax^2 + bx + c $ experiences the factors A (1/2), B (3/7) and also C (- 1/1). What are the values of the coefficients $a$, $b$ and also $c$? This is a trouble offered in the area concerning "Systems of Equalities", yet ...
2022-07-25 17:44:03

Proving $\cos \theta = \sin\left(\frac{\pi}{2} - \theta\right)$ for all $\theta$

By drawing a right triangle it is obvious that $\cos \theta = \sin\left(\frac{\pi}{2} - \theta\right)$. I'm trying to prove to myself that this is true for all values of $\theta$ by following the reasoning on where the sine graph is inverted vertically by plugging in $-\theta$ and then shifted to the right by $\frac{\pi}{2}$ to get the cosine g...
2022-07-25 17:42:42

Sum of digits of numbers

Suppose $N = 1+11+101+1001+10001+\dots+1\underbrace{00\dots00}_{50\text{ zeroes}}1$. When $N$ is written as a single integer, i.e. all terms added, what is the sum of the digits of $N$? I tried subtracting $1$ from each term to get: $$0+10+100+\dots\;,$$ therefore ending with a sum of $\underbrace{111\dots111}_{50\text{ ones}}$. Then I added $50...
2022-07-25 17:21:42

Identifying Asymptotes of a Hyperbola

How would certainly I locate the upright and also straight asymptotes of a $y = \frac{1}{x}$ function algebraically? As an example, $y = -\frac{2}{x+3}-1$ (as you would certainly type right into a calculator). Merely, just how do I locate the x and also y values by considering this formula? To put it simply, the center factor where the asymptote...
2022-07-25 17:20:25

Root equation - What am I missing?

There is a trouble of which I recognize the remedy yet not the addressing process: $(\sqrt{x} + 7)(\sqrt{x} - 1) = \frac{105}{4}$I'm persuaded that up to: $x + 6\sqrt{x} - 7 = \frac{105}{4}$everything is proper. Yet after that, I never ever appear to be able to reach the remedy of $x = \frac{49}{4}$ How can this formula be addressed?
2022-07-25 17:19:26

When the second hand bisects the angle between the hour hand and the minute hand

Find the exact time between $2.25$ pm and $2.26$ pm when the second hand bisects the angle between the hour hand and the minute hand. I mention below my solution to this question. I would like to know whether my solution is correct and also are there any other ways to solve this question. As the second hand bisects the angle formed by the hour a...
2022-07-25 17:17:29

Solving $a = b^2 + 2b^2(1 - b)$ for $b$

My algebra is very rusty, it's been about 15years since I studied, and I was stumped recently when trying to rearrange this formula; $$a = b^2 + 2b^2(1 - b)$$ to give $b$ in terms of $a$. Can someone show me step by step working please :) I remember 'change side, change sign' but it all gets very confusing very quickly. Thanks! So, $$2b^2(1-b)...
2022-07-25 17:16:37

Word problem about travel time on a river with a current

On a river there is no existing from A to B, yet an existing from B to C. A male rows down a stream from A to C in 3 humans resources and also C to A in 7/2 humans resources ; had actually there coincided existing in all the means as from B to C his trip down stream would certainly have taken 11/4 humans resources ; locate the moment his return ...
2022-07-25 17:16:11

Signs in binomial expansions

Edit the title as appears fit. $$\begin{align} (a^3+b^3) &= (a+b)(a^2 -ab+b^2) \\ &= (a+b)^3 -3ab(a+b) \end{align}$$ And so on etc. Now, I just require these developments in addressing square formulas. Yet why do indicators differ in the developments? (asterisk). What controls this? I see that something comparable can be ...
2022-07-25 17:14:07

What kind of graph would $|x-h| + |y-k| = 1$ give?

I recognize that $(x - h)^2 + (y - k)^2 = 1$ is a circle, yet what would certainly the chart resemble for $|x-h| + |y-k| = 1$ and also why would certainly it resemble that?
2022-07-25 17:10:20

Bounding Summations

I have a test showing up that I'm researching for and also I'm rather puzzled by a pair troubles. I require to acquire an asymptotically limited bound for the adhering to summation: Assume $r$ and also $s$ are constants such that $r,s\geq 0$. $$f(x)=\sum_{k=1}^nk^r\log^s(k)$$ So much I have actually thought of $$0\leq f(x)\leq(\sum_{k=1}^nn^...
2022-07-25 17:00:50

Finding Asymptotic Bounds

I do not recognize just how to locate limited (also known as asymptotic) bounds for a function. Take into consideration the function $$f(n)=\sum_{k=1}^nk^r$$ How would certainly I locate limited upper/lower bounds for this. Please aid: - (
2022-07-25 17:00:20

Quadratic equation

Suppose that $a+b = 2m_{1}$ and also $ab = 4m_{1}^{2}-3m_{2}$. Why is the square formula $$y^{2}-2m_{1}y+(4m_{1}^{2}-3m_{2})=0$$ as opposed to $$y^{2}+2m_{1}y+(4m_{1}^{2}-3m_{2})=0$$ In various other words, why is it $-2m_{1}y$ as opposed to $2m_{1}y$ in the 2nd term?
2022-07-25 16:59:58

Does a quadratic necessarily have a root in this interval?

If F (x) is the square $ax^2+bx+c$ with $ac>0$ $b^2-4ac>0$, it holds true that within the interval $[-\frac{b}{a},+\frac{b}{a}]$ there exists a factor $x$ where $F(x)=0$. I was informed this earlier yet I do not see just how that is always real.
2022-07-25 16:58:53

System of equations: $x^2+y=7, y^2+x=11$

Possible Duplicate: ¢ Steps to solve this system of equations During the trip from Moscow to Yerevan my next-door neighbor offered me the adhering to problem: Solve the system: $$\left\{\begin{array}{c}x^2+y=7 \\ y^2+x=11. \end{array}\right.$$ It is very easy to locate 1 of the 4 remedies. Exists an attractive means to locate the various...
2022-07-25 16:57:13

Relationship between two absolute value curves

Describe the partnership in between the contours $|x| + |y| = 1$ and also $|x| + |y-a| =1$, where $a>0$ is a constant.
2022-07-25 16:41:42