0
votes
2answers
111 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
18 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 ...
2
votes
2answers
30 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
35 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
69 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
48 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
33 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
102 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
116 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 ...
1
vote
2answers
170 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
49 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
34 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
46 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
29 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
53 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
77 views

Solve …

This is what I did Can anyone tell me what's wrong me or the question?
0
votes
4answers
75 views

Find $\frac{a^3}{a^6 + 1}$ given a is a root of a quadratic equation

My question is: If a is a root of the equation $x^2 - 3x + 1 = 0$, then find the value of $\frac{a^3}{a^6 + 1}$. So, I figured we can use the Sridharacharya ...
0
votes
1answer
93 views

9 rectangles have the same area as 20 squares

This is a fun little question that I encountered on a problem solving assessment: A small area is covered by 20 identical square tiles or 9 identical rectangular tiles. The length of the side of ...
2
votes
2answers
62 views

How to solve the following pair of equation.

The pair of equation I need to solve is $x^2+12x+y^2-4y=24$ $x^2-6x+y^2+8y=25$ I have no idea on how to do these kinds of problems (may be by elimination?)
1
vote
0answers
68 views

Quadratic equations with prime coefficients

I recently decided to go through old high school notebooks and I found something marginally interesting. I used to note down all kinds of things I came across, and I thought this might be useful for ...
-3
votes
1answer
89 views

To find ? in equation $a^2+?^2=c$

How can we solve for $?$ in the below given Equation: $$a^2+?^2=c$$ I donĀ§t want to use Square or Square root as the the number can be in decimals.
3
votes
5answers
429 views

Condition for a common root in two given quadratic equations

If $a,\;b,\;c$ are in Geometric Progression, then the equations $ax^2+2bx+c=0$ and $dx^2+2ex+f=0$ have a common root if $\;\displaystyle\frac da,\;\frac eb,\;\frac fc$ are in: Arithmetic Progression ...
0
votes
3answers
51 views

Solving an equation involving $x^2$

I have come to a question with the equation: $$6 = x^2 -7x + 6.$$ The answer is $7$. How do I do I find the solution to a problem involving $x^2$?
0
votes
3answers
176 views

Solving a Quadratic Equation “Using a Table and a Graph”

I need to find $a\in \Bbb Z, 0\le a\lt10 : f(1 + \frac{a}{10}) = 0$ for a number of different quadratic functions, for example $f(x) = -x^2 + 4x - 3$, by "using a table and a graph". Can someone ...
0
votes
5answers
61 views

$x^2+y^2=1, 5x+12y+13=0$ Simultaneous Equations

Can someone solve this for me and show working out? I just can't do it and I don't know why I am getting x and y wrong. It will be very much appreciated. As basic as possible as well please.
0
votes
1answer
50 views

How can I solve $\frac{2x}{\sqrt{1-x^2}}=0$

This is what I could come up with: $\dfrac{2x}{\sqrt{1-x^2}}=0$ $\left(\dfrac{2x}{\sqrt{1-x^2}}\right)^2=0^2$ $\dfrac{4x^2}{1-x^2}=0$ I can't go forward from this point because of that stupid ...
1
vote
1answer
39 views

“Alternative factorising method” for quadratics not working

In class my teacher showed us an alternative method for factorising quadratics which are more awkward (i.e. the $a$ in $ax^2+bx+c$ is greater than 1). The method is: 1. Take your quadratic (e.g. ...
6
votes
3answers
279 views

A new way of solving cubics?

I found this (from http://www.quora.com/Mathematics/What-are-some-interesting-lesser-known-uses-of-the-quadratic-formula): So my question is: Can this be generalized to solve any depressed cubic ...
0
votes
1answer
19 views

absolute value in a quadratic

If $a<-2$ is a real number, then the equation: $x^2+a|x|+1=0$ has how many real roots? After finding the roots in terms of $a$, how do I proceed?
0
votes
1answer
52 views

Algebra formulas: answer is positive, but in calculator it's negative.

$$-X^2 + 11X - 30 = 0 $$ $$\frac{-11 + \sqrt{11^2 -4 * 1*30}}{2*1} => \frac{-11 + \sqrt{1}}{2} => -5$$ Why do I get minus? In the book, it shows 5, not -5?
-5
votes
4answers
159 views

Roots of $x^2+3x+2=0$ are infinite !!! [closed]

I have quadratic equation here: $x^2+3x+2=0$ so $(x+2)(x+1)=0$ and I can do $(x+2)=0/(x+1)$ and that solution of the equation is $x+2=0$ so $x=-2$ but my teacher said that it is wrong why? ...
2
votes
3answers
95 views

Prove that for real numbers $x$, if $x^2 - 5x + 4 \ge 0$, then either $x \le 1$ or $x \ge 4$.

Its another homework question that I'm having trouble understanding. The full question is write a detailed structured proof that uses a proof by cases to prove that for real numbers $x$, if $x^2 - 5x ...
4
votes
2answers
120 views

Solving system of multivariable 2nd-degree polynomials

How would you go about solving a problem such as: \begin{matrix} { x }^{ 2 }+3xy-9=0 \quad(1)\\ 2{ y }^{ 2 }-4xy+5=0 \quad(2) \end{matrix} where $(x,y)\in\mathbb{C}^{2}$. More generally, how would ...
1
vote
1answer
75 views

Why does completing the square give you the minimum point?

Say we have an equation:$y=$ ${x^2} + 2x + 1$ Completing the square we get: $\eqalign{ & y={x^2} + 2x + 1 \cr & = {(x + 1)^2} - {(1)^2} + 1 \cr & = {(x + 1)^2} \cr} $ The ...
5
votes
1answer
228 views

Finding integral roots of $x^2 + px + q = 0$ if $p+q=198$.

Given the relation that $p+q=198$, the question is to find all the integral roots of the equation: $$ x^2+px + q = 0 $$ How to proceed? I know we'll have to use Vieta's formulas, but I don't know ...
1
vote
2answers
552 views

coefficients of quadratic function?

In a quadratic function: coefficient $a$ controls the speed of increase/decrease from the vertex. coefficient $b$ controls the downward slope as the function crosses the y-axis. I don't really ...
1
vote
4answers
69 views

Find the roots of the given equation : $2^{x+2}.3^{\frac{3x}{x-1}} =9$ - Logarithm problem

Find the roots of the given equation : $2^{x+2}.3^{\frac{3x}{x-1}} =9$ My working : Taking log on both sides we get : $$\log (2^{x+2}.3^{\frac{3x}{x-1}}) =\log 3^2 \Rightarrow (x+2)(\log2) + ...
3
votes
2answers
775 views

Finding the discriminant and roots of a polynomial

How is the discriminant of a polynomial determined? I know that for a quadratic function, the roots (where $f(x)=0$) are found by $$x=\frac{-b\pm\sqrt{\Delta}}{2a}$$ and here $\Delta$ is the ...
1
vote
2answers
106 views

Can we refer to the standard form of a quadratic equation as the general form as well?

I would like to know if we can refer to $$ax^2+bx+c=0$$ as the "general form" of a quadratic equation, or is it only called the standard form?
1
vote
2answers
57 views

How do you know which substitutions to make to cancel out a term?

I am doing problem B45 from Ivan Niven's "Maxima and Minima Without Calculus" which says: "Consider the quadratic polynomial $f(x, y)=ax^2+2bxy+cy^2+dx+cy+k$ , where the coefficients are real ...
1
vote
1answer
237 views

Solving inequalities, simplifying radicals, and factoring. (Pre calculus)

(Q.1) Solve for $x$ in $x^3 - 5x > 4x^2$ its a question in pre calculus for dummies workbook, chapter 2. The answer says: then factor the quadratic: $x(x-5)(x+1)>0$. Set your factors equal to ...
0
votes
2answers
57 views

Quadratic topic- very basic

Express $x(4-x)$ as the difference of two squares. I do not really quite sure what is meant by difference of two squares.
-1
votes
1answer
79 views

If $a,b,c \in R$ such that $c \neq0$ If $x_1$ is a root of $a^2x^2+bx+c=0, x_2$ is a root of $a^2x^2-bx-c=0 $ and $x_1 > x_2 >0$…

Problem : If $a,b,c \in R$ such that $c \neq0$ If $x_1$ is a root of $a^2x^2+bx+c=0, x_2$ is a root of $a^2x^2-bx-c=0 $ and $x_1 > x_2 >0$ then the equation $a^2x^2+2bx+2c=0$ has roots $x_3$ ...
1
vote
2answers
96 views

System of quadratic equations

How would you solve the following system of equations: $$ x^2 + y = 4 \\ x + y^2 = 10 $$ Thanks very much! I tried defining y in terms of x and then inserting in to the second equation: $$ y = 4 - ...
0
votes
0answers
53 views

If $a,b,c,k \in R, k >0$ and equation $px^2+qx+r=0$ has two real roots $\alpha , \beta $ such that $\alpha < -k$ and $\beta >k$…

Problem : If $a,b,c,k \in R, k >0$ and equation $px^2+qx+r=0$ has two real roots $\alpha , \beta $ such that $\alpha < -k$ and $\beta >k$, then $\forall \beta$ such that $ x \in R ,$ then ...
0
votes
1answer
93 views

Determine all the values of the parameter $a$ for which the inequality $3-|x-a|>x^2$ is satisfied by at least one negative $x$.

I wanted to know, how can I determine all the values of the parameter $a$ for which the inequality $3 - |x-a| > x^2$ is satisfied by at least one negative $x$. I tried for $x<a, |x-a|=-(x-a)$ ...
2
votes
3answers
196 views

Given that the roots of the quadratic equation $x^2+2ax+3a=0$ lie between $-1$ and $1$, what are the possible values of $a$?

In the equation $x^2+2ax+3a=0$ has two solutions $\alpha$ and $\beta$ where $-1<\alpha,\beta<1$. Find out the range of $a$. I tried to solve it by taking $\alpha^2+\beta^2$. But it does ...
2
votes
2answers
83 views

maximum using completing the square

Is it just me, or this problem does sound weird? The Parks Department is fencing a rectangular dog-run (a place for dogs to exercise) in a local park. It will be 7 yards longer than 5 times its ...
0
votes
1answer
57 views

Find the integer solutions

What are the pairs $(A,N)$ where $A,N$ are integers such that the following equation is satisfied: $\large A=\frac{-6+\sqrt{144-12N^2}}{6}$ I know that we should have: $k^2=144-12N^2$ for some ...
3
votes
3answers
137 views

Systems of Quadratic Equations Question

looking for help on this question. Solve the following systems of equations algebraically using the quadratic formula. $$\begin{align} y& =-x^2+2x+9\\ y& =-5x^2+10x+12\end{align}$$ Any help ...