Mathematics Stack Exchange
Mathematics Stack Exchange

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 $$x = \frac{-b\pm \sqrt{b^2-4ac}}{2a}$$

5400 questions
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How to get the summation of a quadratic sequence with O(1)

Is there a formula for cumulative or summation of terms in a quadratic equation? I need an O(1) formula since I need to put this into a code. Thanks. Here's snapshot of my spreadsheet. Currently, I'm just bruteforcing the cumulative value.
Sylpheed
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Is order matter when writing the roots of a quadratic equation?

Equation: $x^2-x-6=0$ The two roots of this equation are $3$ and $-2$. When writing the answer can I also write it as $-2, 3$ or do I have to maintain a certain order?
Russell
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Common zero of two quadratics

Consider$ f(x) =x^2 + ax+b$ , and $g(x) = x^2 +bx+a$ , given that both have one common zero, what is the value of a+b, given $ a\neq b$ Solution according to book: f(1)= g(1)=0 and, hence, a+b=-1 But this doesn't make sense to me, as they could…
tryst with freedom
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How can I solve this system of equation?

I ham given the following problem to solve: 1.9. The program should take three numbers: a; b; c and find the roots of the quadratic equation in the form: If the value of the determinant of the quadratic equation is negative (i.e. ∆ <0), the…
user366312
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How to determine $x$ in $x^2+3x+7=4$?

I'm helping my child with his homework. One of the problems is this: Determine the solution set of the equation $x^2+3x+7=4$ Here is my attempt to determine $x_1$ and…
Glory to Russia
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Let $a,b,c$ be real numbers, $a\ne0$. If a is a root of $a^2x^2++bx+c=0$, $\beta$ is the root of $a^2x^2-bx-c=0$ and $0<\alpha<\beta$ [continued]

Let $a,b,c$ be real numbers, $a\ne0$. If a is a root of $a^2x^2++bx+c=0$, $\beta$ is the root of $a^2x^2-bx-c=0$ abd $0<\alpha<\beta$, then the equation $a^2x^2+2bx+2c=0$ has a root $\gamma$ that always satisfies. A)…
prat
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If the system of inequalities $3x^2+2x-1<0$ and $(3a-2)x-a^2x+2<0$ possesses a solution, find the least natural number $a$

If the system of equations $3x^2+2x-1<0$ and $(3a-2)x-a^2x+2<0$ possesses a solution, find the least natural number $a$ My attempt is as…
user3290550
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If $(x^2-5x+4)(y^2+y+1)<2y$ for all real $y$, then $x$ belongs to the interval $(2,b)$, then $b$ can be?

If $(x^2-5x+4)(y^2+y+1)<2y$ for all real $y$, then $x$ belongs to the interval $(2,b)$, then $b$ can be? $$y^2(x^2-5x+4)+y(x^2-5x+2)+(x^2-5x+4)<0$$ As it is true for all real y, hence $D<0$ $$(x^2-5x+2)^2-4(x^2-5x+4)^2<0$$ Let…
prat
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Let $x_1$ and $x_2$ be the real root of the equation $x^2-kx+(k^2+7k+15)=0$, if the maximum value of $(x_1^2+x_2^2)=\dfrac{18}{x}$, then find 'x'

Let $x_1$ and $x_2$ be the real root of the equation $x^2-kx+(k^2+7k+15)=0$, if the maximum value of $(x_1^2+x_2^2)=\dfrac{18}{x}$, then find the value of $x$? My attempt is as…
user3290550
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Let $\alpha$ and $\beta$ be the roots of $x^2-x-1=0$. If $P_k=(\alpha)^k+(\beta)^k$, $k\ge 1$, then prove that-

a) $P_5=11$$ b) $P_1+P_2+P_3+P_4+P_5 =26$ For the first part $$\alpha^5+\beta ^5$$ $$=(\alpha^3+\beta ^3)^2-2(\alpha \beta )^3$$ I found the value of $\alpha^3+\beta^3=4$ So $$16-2(-1)=18$$ which doesn’t match. In the second part depends on the…
Aditya
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If both roots of the equation $ax^2-2bx+5=0$ are $\alpha$ and roots of the equation $x^2-2bx-10=0$ are $\alpha$ and $\beta$.

find $\alpha^2+\beta^2$ Both equations have a common root $$(-10a-5)^2=(-2ab+2b)(20b+10b)$$ $$25+100a^2+100a=60b^2(1-a)$$ Also since the first equation has equal roots $$4b^2-20a= 0$$ $$b^2=5a$$ I could substitute the value of a in the above…
Aditya
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Interpreting relationship between points on a quadratic curve

Taking a quadratic $Q(x)=ax^2+bx+c$, we can rearrange it to $$\frac{Q(x)-c}{x}=ax+b$$ Then, supposing there are two points $(x_1,y_1),(x_2,y_2)\in Q$, we have: $$a=\frac{\frac{y_1-c}{x_1}-\frac{y_2-c}{x_2}}{x_1-x_2}$$ Bringing in a third point,…
Rhys Hughes
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Find factor $f(x)$ conmmon to two quartic equation

Let $f(x) = x^2 + bx + c$, where $b, c ∈ R$. If f(x) is a factor of both $x^4 + 6x^2 + 25$ and $3x^4 + 4x^2 + 28x + 5$, then find $f(x)$ My approach ,on dividing both quartic equation by $f(x)$ remainder is zero, but not getting the answer
Samar Imam Zaidi
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If $\alpha$ and $\beta$ are roots of the equation $x^2-2x+2=0$ then the least value of n for which $\left(\frac{\alpha}{\beta}\right)^n=1$

So simple thought here. Shoudnt n simply be zero? I mean there is no condition that states n cannot 0, so why is the answer 4? It may be an obvious answer, but i can’t get my head over it. Thanks!
Aditya
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How should I solve the equation $\sqrt{x-\frac 1x}+\sqrt{1-\frac 1x}=x$

I could square both sides of the equation, but that ends up giving me a cubic to solve. What I need is a beginning approach to solve such questions, not the whole answer. Thanks
Aditya
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