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Questions tagged [quadratics]

Questions about quadratic functions and equations, second degree polynomials usually in the forms $y=ax^2+bx+c$, $y=a(x-b)^2+c$ or $y=a(x+b)(x+c)$.

Questions about quadratic functions and equations, second degree polynomials usually in the forms $y=ax^2+bx+c$, $y=a(x-b)^2+c$ or $y=a(x+b)(x+c)$.

The root of $y=ax^2+bx+c$ can be solved by the formula $$x = \frac{-b\pm \sqrt{b^2-4ac}}{2a}$$

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Solve quadratic equation expressed as sum of quadratic terms

Let's say I have the following quadratic equation $$ (x - a_1)^2 + (x - a_2)^2 + \dots + (x - a_n)^2 = b $$ This will have $2$ solutions (complex solutions are valid). One could expand that out, collect the terms and then use the quadratic formula. …
Makogan
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Formation of two families of quadratic equations

Question 1: Find all values of $a,b$ such that roots of $x^2+ax+b=0$ be of the type $(p,p^2)$. There can be infinitely many pairs of $(a,b)$ such as $a=-(p+p^2),b=p^3$. One may also eliminate $p$ from these two equations: $b^{2/3}+b^{1/3}=-a$…
Z Ahmed
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Quadratic Word Problem About The Width Of A Basketball Court

I was solving this problem from my algebra textbook: A basketball court measures $25m \times 15m$. The court is surrounded by a row of benches that is the same width on all sides. If the row of benches has total area of $325m$$^2$, find the…
Jaireet Chahal
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This question is asking about the number of quadratic polynomials possible under given conditions

Find the number of quadratic polynomials ax²+bx+c ,which satisfy the following conditions : (I) a,b,c are distinct (II) a,b,c ∈ {1,2,3,4......999} (III) (x+1) divides (ax²+bx+c) Solution by me: Since all of coefficients lie between 999 and 1 I wrote…
Vignesh
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The meaning of the coefficients in a quadratic polynomial

Let's say we are given a quadratic polynomial $ax^2+bx+c$. Without the use of derivatives, what is the meaning of the coefficients when it comes to graphing the polynomial? The coefficient $c$ determines the intersection with the y-axis by setting…
TJ123
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Is there a simple way to find the integer part (floor) of the positive root of a quadratic equation?

In order to show the positive root of a quadratic equation in Simple Continued Fractions I map the quadratic equation like; $cx^2+(d−a)x−b=0\implies x=\frac{ax+b}{cx+d}$ where $a > c$ or perhaps $a>d$. I believe this tells me that the positive root…
Redu
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Product of real quadratic roots

If product of roots '$p$' of equation $x^2 – 2ax + 8 – a = 0$ lies between its roots, then maximum integral value of '$p$' is _____ My approach is as follow $f\left( x \right) = {x^2} - 2ax + 8 - a$ ${D^2} = 4{a^2} - 4\left( {8 - a} \right) \ge 0…
Samar Imam Zaidi
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The equation $y = x^2 + \frac{1}{2}ax + a$ represents a parabola for all real values of $a$

The task is to show that any parabola with the value a passes through shared point. I graphed the parabola and understand that the shared point is $(-2,4)$. My current issue is generalizing why this is and how to prove it.
Jake Lazard
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Sign of $P(Δ)$ with $P(x)=ax²+bx+c$ and $Δ=b²-4ac$

Let $P(x)=ax^2+bx+c$ be a polynomial with $a≥0$ and $b≥\frac{1}{8a}$, and $Δ=b^2-4ac$. Show that $P(Δ)≥0$. My answer: We have $Δ=b^2-4ac$. If$Δ≤0$, then $P(Δ)≥0$. If $Δ>0$, there are two cases: If $c≥0$, $P(Δ)=aΔ^2+bΔ+c≥0$. If $c<0,Δ=b^2-4ac$ then…
connaissant
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Two quadratic equations having a common root

If quadratic equations $a_1x^2 + b_1x + c_1 = 0$ and $a_2x^2 + b_2x + c_2 = 0$ have both their roots common then they satisy, $$\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$$ But even if both the quadratic equations have only one common root…
marks_404
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Find the quadratic equation from given relatioship

the quadratic equation whose roots are a and b where $a^2 +b^2=5$ and $3(a^5+b^5)=11(a^3+b^3)$ What I Tried $a^2 +b^2=5$ $(a+b)^2-2ab=5$ $(\text{sum of roots})^2 -2(\text{products of roots})=5$ $3(a^5+b^5)=11(a^3+b^3)$ …
Rishav Bhagat
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Finding the Equation of Parabola

Write the equation in the form $y=a(x-h)^{2}+k$ with zeros -4 and 8, and an optimal value of 18. I'm not sure what "optimal value" means first of all- I think it means that the maximum value has a y-value of 18. What I've done so…
Sujaan Kunalan
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if i drive a car 60 miles per hour on a full tank, it'll run for 5 hours. for every mile per hour i speed up, the car will run 10 min less.

if I drive a car 60 miles per hour on a full tank, it'll run for 5 hours. for every mile per hour I speed up, the car will run 10 min less. for every mile per hour I slow down the car will run 10 min more. a)in terms of x how far can I drive on one…
allan
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System of two quadratics equation, $P(x)$ and $Q(x)$

If $P(x) = ax^2 + bx + c$ and $Q(x) = – ax^2 + dx + c$, $ac \ne 0$, then the equation $P(x) . Q(x) = 0$ has (A) Exactly two real roots (B) At least two real roots (C) Exactly four real roots (D) No real roots My approach is as follow Let…
Samar Imam Zaidi
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What does the quadratic formula really do?

I am trying to factorize an expression in terms of x: $3x^2 + 8x + 2$. But I know that in order to do that I have to make it equal to zero. If I find the roots of that equation then equal to zero using the quadratic formula I obtain $2$ surds :…
butterfly
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