Questions on quadratic equations, second degree polynomials usually in the forms $y=ax^2+bx+c$, $y=a(x-h)^2+k$ or $y=a(bx+c)(cx+d)$.

learn more… | top users | synonyms

0
votes
2answers
90 views

Solving for $a$ and $b$

How would you solve the equality: $$a\,b\,\left(b+a\right)=1$$ in terms of a and/or b? Would you subtract 1 from both sides and work from there? Or would you simply expand and work from there?
0
votes
2answers
39 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 ...
4
votes
2answers
73 views

Integral values of an expression

Let $b=\sqrt{a^2+5a+8}-\sqrt{a^2-3a+4}$ Find number of integral values of b. My $long$ way using Calculus : Find domain of function : $R$ Note that function is continuous Prove the function is ...
0
votes
2answers
36 views

What does each term in the trinomial represent and their relation to each other? [duplicate]

I am trying to get a grasp or concept understanding what how trinomials answer questions other than answering questions in an algebra class. I. Looking for the practical application. What does the ...
0
votes
1answer
32 views

With Trinomial's, can someone explain the purpose of the first, second and third term. In layman terms.

I know that the first term is a quadratic and I suppose that lets us know we are dealing with identifying a curve, and the third term is our constant. I just can't quite put it all together as how ...
2
votes
2answers
49 views

Find the range of values of $k$ for which the following equation has real roots.

Find the range of values of $k$ for which the following equation $x^2+(1-k)x=k$ has real roots. I tried it, for real roots, $b^2-4ac \geqslant 0$ $x^2+(1-k)x-k=0$ $(1-k)^2-4*1*(-k)\geqslant0$ ...
19
votes
10answers
3k views

Why is every equation always equal to zero? [closed]

A linear equation is $$ ax + b = 0 ; \,\, \,\, a\neq 0 $$ A quadratic equation is $$ax^2 + bx + c = 0 ; \,\, a\neq 0 $$ And so on... Why are all equal to zero? Why have mathematicians defined it ...
0
votes
2answers
110 views

Solving quadratics using $y=a(x-p)^2+q$ ?

The vertex is $(-4, 2)$ and the y-int is $(-3,-1)$. How would I solve this, or find out what "$a$" is so I can write the equation and graph it?
1
vote
3answers
126 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
56 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 ...
1
vote
5answers
34 views

Linear systems from positions a quadratic passes through?

I don't understand the question. $$y = ax^2 + bx + c$$ passes through the points $(1,-4),(-1,0),(2,3)$. Write down a linear system (unknowns $a,b,c$) of three equations relating the unknowns to each ...
0
votes
5answers
114 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
384 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
6k 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
93 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
123 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
54 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
41 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: $b^2-4ac=0$ ...
0
votes
2answers
27 views

all the values of c for the equation: x^2-5x+c>2

We need to find all the values of c for this equation: x^2-5x+c>2 This question was on my exam I didn't know how to get started on the question. Could some one help me out.
1
vote
2answers
200 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?
0
votes
3answers
93 views

Solving a quadratic using vietas theorem I keep going in circles.

I am trying to solve the quadratic equation $x^2-48x+432=0$ with out directly factoring OR using the quadratic equation. I am going with vieta. So $$r+s=48$$ and $$rs=432$$ I've already solved it by ...
0
votes
0answers
42 views

Solving quartic equations

I am solving quartic equations by finding the intersections of two quadratics. The given quartic function is of the form $x^4 + px^2 + qx + r = 0$. I know one of the quadratic functions to be $y=x^2$. ...
0
votes
1answer
53 views

How to turn algebra into quadratic equation? [closed]

I have to find the maximum value of $x^2+(5+x)^2$ using a parabola. How do I turn the $x^2+(5+x)^2$ into a quadratic equation?
1
vote
1answer
79 views

Solve …

This is what I did Can anyone tell me what's wrong me or the question?
0
votes
4answers
90 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 ...
3
votes
3answers
132 views

Finding maximum value for a function

I was working on this question to find the following function's maximum value.Let $$y=f(x)={{(\sqrt{-3+4x-x^2}+4)}}^2 + (x-5)^2$$ where $$1 \le x \le 3$$.I have to find it's maximum value. I tried by ...
3
votes
1answer
36 views

Problem with system of equations

I wonder how to solve this system of equations: $\begin{cases} 2x^2+y^2=43\\2x^2+4xy=78\end{cases}$ when I subtract I have $y(4x-y)=35$ but I don't if it is good way to look for the solutions.
-1
votes
1answer
53 views

Finding the rule of a quadratic graph

I am trying to find the rule. Insofar: $y = a(x-b)^2 + c$ Turning point is $ (1,9) $ So $ b = 1 $ and $ c = 9 $ $y = a(-4-1)^2 + 9$ $-16 = a(-4-1)^2 + 9 $ $-25 = a (-4-1)^2 $ $-25 = a (-5)^2 ...
0
votes
2answers
102 views

Reducing quartic equations to quadratic

I'm trying to re-learn basic math/algebra, and I can't get passed one question concerned with reducing quartic equation to a quadratic: Find, correct to 3 significant figures, all the roots of the ...
0
votes
1answer
1k views

Solving a distance/time/speed problem using the quadratic formula. [closed]

"The distance between Toronto and Ottawa is 352.72 km. The speed on a road trip from Ottawa to Toronto was double of the return, and therefore the drive took 2 hours less. What was the speed on the ...
0
votes
2answers
80 views

Rearranging quadratic equations

I have an equation I'd like to rearrange that I'm having trouble with: The equation goes: $$r = 2ut - 0.333(2ut)^2$$ (on a computer I enter ...
0
votes
1answer
94 views

Solve a bit tricky system of equations

I want to solve the system for $x$, $y$ and $z$. Is there any smart trick to solve it? $$\begin{cases} 2a(ax+by)+2c(cx+dy)+2zx=0 \\ 2b(ax+by)+2d(cx+dy)+2zy=0 \\ x^2+y^2-1=0\end{cases}$$ $a,b,c,d \in ...
0
votes
1answer
134 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 ...
3
votes
1answer
84 views

Find the range of values of $p$ if $(\cos p -1)x^{2}+(\cos p)x+\sin p =0$ has real roots in the variable $x$.

Find the range of values of $p$ if $(\cos p -1)x^{2}+(\cos p)x+\sin p =0$ has real roots in the variable $x$. Restrict the values of $p$ in $[0,2\pi]$. The given equation has real roots if: $$\cos^2 ...
-2
votes
3answers
57 views

A question on quadratic equations.. Given below in the picture.

PLease also tell how u got to the answer as I want to know the way to solve further questions
2
votes
2answers
64 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
2answers
88 views

If $ax^2+bx+c=0$ and $2x^2 +3x+4=0$ have a common root where $a,b,c \in \Bbb N$,find least value of $a+b+c$

problem :If $ax^2+bx+c=0$ and $2x^2 +3x+4=0$ have a common root where $a,b,c \in \Bbb N$,find least value of $a+b+c$ Solution: Here $2x^2 +3x+4=0$ will give complex roots These roots will ...
1
vote
1answer
22 views

Consider the following simultaneous equations in $x$ and $y$…where $a$ is a real constant: $x+y+axy=a$,$x-2y-xy^2$

Consider the following simultaneous equations in $x$ and $y$: $$x+y+axy=a$$ $$x-2y-xy^2=0$$ where $a$ is a real constant. Show that these equations admit real solutions in $x$ and $y$. I could not ...
1
vote
2answers
43 views

Three variable systems if equations.

Given the quadratic function $y=x^2 + 4$ and the linear function $y=x + b$, determine all the possible values of $b$ that would result in a system if equations with two solutions, exactly one ...
-3
votes
1answer
98 views

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

How can we solve for $?$ in the below given Equation: $$a^2+?^2=c$$ I don§t want to use Square or Square root as the the number can be in decimals.
0
votes
2answers
89 views

Complex numbers - Quadratic formula?

Let a and b be real numbers. The complex number 4 - 5i is a root of the quadratic $z^2 + (a + 8i) z + (-39 + bi) = 0$. What is the other root? I did a lot of work on hand and plugging this into the ...
0
votes
2answers
92 views

Query about a statement on the consequence of two quadratic equations having a common root

I have read an answer (in this site, but I lost the question number) saying something like the following:- If the quadratic equations F(x) = 0 and f(x) = 0 have a common root, then the quadratics are ...
3
votes
5answers
2k 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
2answers
34 views

Find quadratic equation based on 2 tangents

I would like to know a way to find an quadratic equation that had 2 given tangents: For example here is 2 tangents equations: y = 1/2 x y = 2 x + 2 and 2 abscisses x = 0 x = 3 Is there a ...
0
votes
3answers
53 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
597 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
1answer
37 views

information content of a quadratic surd

how much information is required to construct the equation: $$ X^2 - 2=0 \; ? $$ suppose, in a spirit of seasonal festivity, we squander a few further bits, and pamper ourselves with the additional ...
0
votes
0answers
61 views

If $3x^{2}-2(a-d)x+(a^{2}+2(b^{2}+c^{2})+d^{2})=2(ab+bc+cd)$, then

If $3x^{2}-2(a-d)x+(a^{2}+2(b^{2}+c^{2})+d^{2})=2(ab+bc+cd)$, then $A.$ a,b,c,d are in G.P. $B.$ a,b,c,d are in H.P. $C.$ a,b,c,d are in A.P. $D.$ None of the above Tried writing the expression as a ...
2
votes
1answer
62 views

how to prove roots quadratics

the quadratic equation $3(k+2)x^2+(k+5)x+k=0$ has real roots show $(k-1)(11k+25) \geq 0 $ If $\Delta$ greater than $0$ it has real roots so, $$\Delta = (k+5)^2 - 4 \cdot (3(k+2))\cdot k$$ ...
0
votes
4answers
73 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.