2
votes
5answers
71 views

Algebraic process to find numbers so that $xy=45$ and $x+y=18$

Can someone help me with the following question? The sum of two numbers is $18$ and their product is $45$. Find the numbers. I know that the answer is $15$ and $3$. But how do I find that answer ...
1
vote
1answer
39 views

Quadratic equations - finding value of x

Question: Solve the equation $$\frac{2x}{3x^2 - x+2} - \frac{7x}{3x^2 + 5x+2} = 1$$ I attempted to split the two quadratic equations into their roots, however, was unable to do so. Then I tried to ...
2
votes
4answers
53 views

Quadratic equation - solving for x

Question: Solve the equation $$(x+2)(x+3)(x+8)(x+12) = 4x^2$$ I tried to solve the equation by expanding the LHS and then equating it to the RHS, but that just doesn't seem to be feasible. I am ...
1
vote
7answers
211 views

How to solve the quadratic equation $x^2-1=2$?

Solve $x^2-1=2$ I have no idea how to do this can somebody please help me? I have tried working it out and I could never get the answer.
2
votes
2answers
56 views

Using factoring to solve the equation $(r^2 + 5r - 24)(r^2 - 3r + 2) = (4r - 10)(r^2 + 5r - 24)$

Solve for all values of $r$: $$(r^2 + 5r - 24)(r^2 - 3r + 2) = (4r - 10)(r^2 + 5r - 24)$$ I'm not sure how my thinking isn't really correct here. I know this all seems very elementary and such, ...
2
votes
4answers
64 views

Show that $ax^2+2hxy+by^2$ is positive definite when $h^2<ab$

The question asks to "show that the condition for $P(x,y)=ax^2+2hxy+by^2$ ($a$,$b$ and $h$ not all zero) to be positive definite is that $h^2<ab$, and that $P(x,y)$ has the same sign as $a$." Now ...
-1
votes
2answers
71 views

How to write these quadratic equation in general form? [closed]

Write the quadratic equation in general form: 1. $x^2=16x$ 2. $13-3(x+7)^2=0$ 3. $x(x+2)=5x^2+1$
-1
votes
2answers
35 views

How to factor the quadratic $6x^2-16x=0$? [closed]

How to factor the quadratic $6x^2-16x=0$ ? I need help solving this. I am aware of how to solve this when it is a polynomial, but I do not know how to solve with only two terms?
0
votes
3answers
56 views

How to solve this word problem on the topic of quadratic equations?

The maths teacher of Mumbai is transferred to another school. The students of Class 10 decided to buy a book for 360 rupees(currency) as a gift for her. On the farewell day, 4 students did not turn up ...
3
votes
2answers
70 views

Solution for this Logarithmic Equation

Recently I was going through a problem from the book Problems in Mathematics - *V Govorov & P Dybov* . $$(x-2)^{\log^2(x-2)+\log(x-2)^5-12}=10^2\log(x-2)$$ I tried solving by first considering ...
1
vote
1answer
69 views

How to solve $\frac{2x+1}{2x-3}+\frac{7x\:}{9-4x^2}=1+\frac{x-4}{2x+3}$ for $x$?

Can somebody explain me this one! $\frac{2x+1}{2x-3}+\frac{7x\:}{9-4x^2}=1+\frac{x-4}{2x+3}$ My book says the answer is $x_1 = 0$; $x_2 = 6$. I tried to solve it and got stuck somewhere in: ...
0
votes
5answers
54 views

Solving a quadratic trigonometric equation?

The equation is $6 \cos^2x+\cos x=1$, My work: $6x^2+x-1=0$ $(3x-1)(2x+1)$ $3x-1=0 ∨ 2x+1=0$ $x=\frac{1}{3} ∨ x= \frac{-1}{2}$ But I do not know how to progress further.
0
votes
3answers
86 views

Using sum/product of quadratic roots to solve cubic equation

Given $\alpha$ and $\beta$ are the roots of the quadratic equation $6x^2 + 2x - 3 = 0$, how do I find the value of: $$ \alpha^3 + \beta^3 $$ and: $$ \frac{1}{\alpha^3} + \frac{1}{\beta^3} $$ ...
0
votes
2answers
35 views

How to prove one of these equations has real roots?

Question: If $\;a,\, b,\, c,\, d \;$ are real and $\;ac = 2( b + d) $ ,then show that at least one of the equations: $\; x^2 + ax + b = 0 \;$ and $\; x^2 + cx + d = 0 \;$ has real roots. I've ...
0
votes
3answers
39 views

Quadratics with unknowns

If $5x^2 – t = 4x$, and $x$ and $t$ are both positive real numbers. What is $x$ equal to? How do you find $x$? Is there a specific formula?
0
votes
1answer
21 views

Quadratic factor to complex numbers

How to convert this quadratic factor to complex number form? (With steps please) Reference: $Z = a + bi$, $i = \sqrt{-1}$ $$-3 + \frac{\sqrt{-12}}{2}$$ Thanks!
0
votes
4answers
39 views

Finding the three unknowns

Can someone show me the steps to finding the three unknowns of these two equations. $$-a-bx+cx^2 = x^2+2x+1$$ The answers are $a=\ ...\ $, $b=\ ...\ $, and $c=\ ...$ , but I can't see how they ...
2
votes
1answer
65 views

Expanding Square Roots, Why No Negative?

I haven't thought through algebra in a while and the last explanation I received of this seemed arbitrary. I hope I can get some clarification here. I understand that $\sqrt{+a} = \pm b$. Here's ...
1
vote
1answer
47 views

Quadratic inequality with parameter

Hi I've got this inequality with parameter $a\in R$ $\frac{x+a}{x}\le x+2$ I've solved it but I've got different results than book. I've done it by dividing it into 2 cases. 1. x<0 2. x>0 and then ...
4
votes
7answers
89 views

How to find $x^2 - x$?

I'm quite a novice when it comes to maths. I'm on a problem in which I have had to isolate $x$ , through factorials which I completed without problem. However, now I am stuck on a seemingly more minor ...
0
votes
1answer
77 views

Find pressure in a sinusoidal function

Tiffany is a model rocket enthusiast. She has been working on a pressurized rocket filled with laughing gas. According to her design, if the atmospheric pressure exerted on the rocket is less than 10 ...
0
votes
2answers
37 views

What are the parameters of a parabola

In the following figure I understand the $bx+c$ part. It is simply the equation of a line. But I don't understand where did $ax^2$ came from? What exactly is it? What does $a$ tell us about a ...
1
vote
2answers
48 views

Find maximum of a system of equations

You have 300 meters of fencing with which to build two enclosures. One will be a square, and the other will be a rectangle where the length of the base is exactly twice the length of the height. (a) ...
6
votes
4answers
128 views

Solve $x^{3}-3x=\sqrt{x+2}$

Solve for real $x$ $$x^{3}-3x=\sqrt{x+2}$$ By inspection, $x=2$ is a root of this equation. So, I squared both sides and divided the six degree polynomial obtained by $x-2$. Then I got a ...
0
votes
5answers
64 views

How do you factor a quadratic expression, without using the formula?

I am asked to factor $2x^2 -3x+1=0 $ using factorization, but I run into fractions, and it becomes very messy and complicated to deal with, especially since specifically asked not to use the formula. ...
7
votes
3answers
116 views

How to solve the following? $ x^3+1=2{(2x-1)}^{1/3} $.

Find all the real solutions of $$x^3+1=2{(2x-1)}^{1/3} $$ I tried to cube both sides but got messed up with a nine degree equation! Please help. Thanks in advance!
3
votes
2answers
73 views

If $P(x) = ax^2 + bx + c$ and $Q(x) = -ax^2 + dx + c$, then prove that $P(x) \cdot Q(x) = 0$ has at least two real roots?

How should i solve the same? I assumed the roots be $ \alpha, \beta $ for $ P(x) $ and $ \gamma, \delta $ for $ Q(x) $. Product of roots turn out to be of the opposite signs, being $$ \alpha \cdot ...
1
vote
1answer
24 views

Find values of the parameter a so that equation has equal roots.

$x^2+2a\sqrt{a^2-3}x+4=0$ My final result was 2 and -0.5. Was it correct?
1
vote
1answer
44 views

how should i go about solving the following problem??

$f(n)=a^n-b^n$ where $a$ and $b$ are roots of the following equation .$$5x^2-2x+1=0$$ Then find the value of $$\frac{5f(10)+f(9)}{f(8)}$$ I realised we can use the 5 in the equation as $\frac{1}{ab}$ ...
5
votes
4answers
103 views

What is the minimum value of $abc$

If the roots of the equation $$ax^2-bx+c=0$$ lie in the interval $(0,1)$, find the minimum possible value of $abc$. Edit: I forgot to mention in the question that $a$, $b$, and $c$ are natural ...
1
vote
2answers
32 views

Exponential Growth Rates

So if you are given two different numbers to determine a growth rate, do you use to largest number compared to the value when x=0. For example the problem I am working on is: Your grandfather ...
4
votes
2answers
64 views

Evaluate $a+b+c+d$

If $a$, $b$, $c$, and $d$ are distinct integers such that $$(x-a)(x-b)(x-c)(x-d)=4$$ has an integral root $r$, what is the value of $a+b+c+d$ in terms of $r$? I tried to analyze graphically by ...
3
votes
2answers
60 views

Find the value of $\left | b-c \right |$

Given that $a, b, c \in \mathbb{Z}$, $a>10$ and $$(x-a)(x-12)+2=(x+b)(x+c)$$ Find the value of $\left | b-c \right |$ NOTE: The answer to this problem (as given on the last page of my book) is ...
5
votes
2answers
143 views

Let $f(x)=ax^2+bx+c$ where $a,b,c$ are real numbers. Suppose $f(-1),f(0),f(1) \in [-1,1]$. Prove that $|f(x)|\le \frac{3}{2}$ for all $x \in [-1,1]$.

Let $f(x)=ax^2+bx+c$ where $a,b,c$ are real numbers. Suppose $f(-1),f(0),f(1) \in [-1,1]$. Prove that $|f(x)|\le \frac{3}{2}$ for all $x \in [-1,1]$. I made quite a few attempts but could not ...
0
votes
2answers
132 views

How would I solve the quadratic $x^2+3x-70=0$?

How would I solve the following quadratic equation $$x^2+3x-70=0 $$ This is my attempt below $$(x-7x) (x+10x)=0 $$ $$ x-7x=0 \implies -6x=0 \implies x=6$$ $$x+10x=0 \implies 11x=0 \implies ...
0
votes
1answer
21 views

Verification of solutions to some polynomial prob/

$\boxed{\text{Problem 1}}$ Find the other solution of the equation $(1+\sqrt3)x^2-(5-\sqrt3)x+6-6\sqrt3=0$ given that $2$ is a solution Ma solution: $x_1\cdot x_2=\tfrac ca$ therefore letting ...
3
votes
2answers
37 views

prove for p(x) which is a quadratic polynomial

$p(x)$ is a quadratic polynomial . Prove that any given number for $a$ with one exception , we can find a number $b$ such that $p(a)=p(b)$ and $a$ is not equal to $b$.
0
votes
2answers
38 views

How to use $t(29/\sqrt{2})<0$ where $t(x)=x^2-41x+420$ to prove that $41/29<\sqrt{2}<42/29$??

So I was investigating different ways to approximate $\sqrt{2}$. Here's my latest: $$Let:t(x)=x^2-41x+420$$ then the roots of $t(x)$ are $20$ and $21$. I showed that then $t(x)=(x-20)(x-21)$ and ...
1
vote
3answers
110 views

Find the range of values of $x$ for which $1-x<(x-1)(5-x)<3$.

Find the range of values of $x$ for which $1-x<(x-1)(5-x)<3$. First of all, I solved $1-x<(x-1)(5-x)<3$ which gives me $(x-1)(x-6)<0$ and $(x-4)(x-2)<0$. How to find the range, ...
1
vote
1answer
54 views

Found an example for solving via quadratic formula in a book where I am wondering if this is correct

As a refresher, I was skimming through a free Calculus online textbook "MOOCULUS massive open online calculus" (https://mooculus.osu.edu/handouts) and stumbled upon the following example solving a ...
-2
votes
2answers
43 views

Find the range of values of $x$ for the inequality $x^2-4x-1>0$ [closed]

Find the range of values of $x$ for the inequality given. $x^2-4x-1>0$
0
votes
5answers
112 views

Show that $3x^2-4x+2$ is always greater than $0$.

How do I show that the function $3x^2-4x+2$ is always greater than $0$?
1
vote
1answer
331 views

Find the maximum or minimum value of the quadratic function by completing the square.

Find the maximum or minimum function of the quadratic function by completing the squares. State the value of $x$ at which the function is maximum or minimum. $y=3x^2+7x+9$ I already posted similar ...
3
votes
2answers
2k views

Find the maximum or minimum value of the quadratic function.

Find the maximum or minimum value of the quadratic function by completing the squares. Also, state the value of $x$ at which the function is maximum or minimum. $y=2x^2-4x+7$ $x^2$ has a coefficient ...
1
vote
4answers
77 views

Convert the L.H.S to the form of the R.H.S by completing the square.

Given that $x^2-3x+5=(x-p)^2+q$ for all values of $x$, calculate the value of $p$ and of $q$. A book example tells me to firstly convert the L.H.S to the form of the R.H.S by completing the square. ...
0
votes
3answers
84 views

Find the value of $p$ and $q$ of the quadratic equation.

The quadratic equation $x^2+px+q=0$ has roots $-2$ and $6$. Find the value of $p$ and $q$. Do I have to make two equations? Something like this? When $x=-2$, (real and distinct roots) ...
0
votes
1answer
51 views

Inequality challenge

I was studying inequations when I encountered this problem here. How can I find a region of values for m where this inequation is true? $$-3<\frac{x^2+mx-2}{x^2-x+1}>2$$ Thanks
1
vote
1answer
35 views

Find the possible values of $p$ for which the equation has coincident roots.

Find the possible values of $p$ for which the equation $(2p+3)x^2+(4p-14)x+16p+1=0$ has coincident roots. Coincident roots means 'equal roots'. For equal roots, we should use: $b^2-4ac=0$ ...
1
vote
2answers
123 views

Find the possible values of $k$, if the equation has equal roots.

The equation $x^2+5k=kx+x+19$ has equal roots. Find the possible values of $k$. Um having problem in rearranging the equation; $x^2+5k-kx-x-19=0$ $x^2+k(5-x)-x-19=0$ What is the next step?
1
vote
1answer
78 views

Solve …

This is what I did Can anyone tell me what's wrong me or the question?