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

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manipulation of quadratic equation from roots

If $\alpha$ and $\beta$ are the two roots of equation of $ax^2 + bx + c$ .How is $$cx^2 + bx + a = c(x-1/\alpha)(x-1/\beta)$$ Later Edit: How to deduce $$c(x-1/\alpha)(x-1/\beta)$$ from $$cx^2 + bx + a$$ , when we dont know if $1/\alpha$ and…
MANVITH LB
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Find the range of $a$ for the following quadratic equation.

$f(x) = x^2 +(a+3)\lvert x \rvert + 4 = 0$ This is the quadratic equation They have given condition that find the range of $a$ for which the roots are real So what I did was $D\geq 0$ Solved the condition and got $(a+7)(a-1) \geq 0$ So $a \in…
Student 4
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finding intersection of a parabola and its shifted version

I was trying to solve this problem for the SAT math section: I was able to guess the answer (5) from the multiple choice options, but I'm not sure what the solution process is. The discussion on the practice site I'm using says you should calculate…
TKR
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Quadratic equation with one of the roots as a part of equation

If $\alpha,\beta$ are roots of $x^2-4\alpha+1=0,$ then equation whose roots are $\frac 1 {4-\alpha}, \frac 1 {4-\beta}$ is? I'm not sure how to go about this. One thing I noticed is that $$\alpha^2 -4\alpha+1=0$$ $$1=\alpha(4-\alpha)$$…
Righter
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Quadratic equation $(\ell-m)x^2-5(\ell+m)x-2(\ell -m) =0 $

If $\ell$, m, n are real,$\ell\ne m$, then the roots by the equation :$(\ell-m)x^2-5(\ell+m)x-2(\ell -m)=0$ are (A)Real and equal (B) Complex (C)Real and Unequal (D) None of these My approach is as follow The discriminant $T = 25{\left( {\ell + m}…
Samar Imam Zaidi
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Let f(x) be a quadratic polynomial satisfying f(2) + f(4) = 0. If unity is one root of f(x) = 0 then find the other root.

Let the root be 1 be $x-1=0$ Let it be called condition 1: $f(2)$ +$ f(4) $= $f(1)$ , not f(x). For a Q like this , I can assume $ax^2$ + bx + c=0 where $a , b , c$ can be negative , +ve or even a bigger value for example ($9$ or$ 9+a$). Q1…
S.M.T
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How to express m in terms of a?

I have to solve this question: The quadratic function f(x) = 2x² + ax - 1 in x satisfies f(-1)≥-3 ,f(2)≥3 Let us consider the minimum value m of f(x) (1) m can be expressed in terms of a as: m = -A/B a² - C (2) The range of the values of a such that…
Science boy
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Proving a quadratic equation having real and positive solutions

How should I go around on proving the quadratic equation $$a^2 x^2 +(2ac-b^2)x+c^2=0$$ having real and positive solutions? I tried to use the fact that if a quadratic equation has real and positive solutions, then the discriminant is greater or…
Loo Soo Yong
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Is $ax^2 + bx = 0$ considered a quadratic equation? Or is it linear, since it simplifies to $ax+b=0$?

I know that a quadratic equation can be represented in the form $$ax^2 + bx + c = 0$$ where $a$ is not equal to $0$, and $a$, $b$, and $c$ are real numbers. However, if there is an equation in the form $$ax^2 + bx = 0$$ would it be classified as a…
Twilight
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Quadratic with non numeric coefficents. Find where q has no real roots

Be patient with me here I'm picking up prep to do my A level maths after not doing any for the last 16 years... My question is for which values of p does the equation f(x)=0 have no real roots px^2 - px + 3x - 4 I'm really struggling with this one,…
Jack douglas
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For what values of $\rho$ is $\lambda \geq 0$ in $(1-\lambda)(\lambda^2 - 2\lambda + 1 - \rho) - 0.5( 0.5 - 0.5\lambda - 0.5\rho)$?

Consider the expression $$ (1-\lambda)(\lambda^2 - 2\lambda + 1 - \rho) - 0.5( 0.5 - 0.5\lambda - 0.5\rho) = 0 $$ We seek the entire range of values of $\rho$ such that $\lambda \geq 0$ in the above expression. Note that the constraints on…
student010101
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$x^2 +2ax+b=0$ where $x_1$ and $x_2$ are real solutions. For which value of $b$ function $f(b)=|x_1-x_2|$ reaches maximum if $|x_1-x_2|=2m$

$x^2 +2ax+b=0$ ; | A-C| = 2m. Roots are A,C-real,distinct. Then , b belongs to ? How I solved it till now : Using formula of A-C I,e ALPHA - beta = $\sqrt{D}$/a $\sqrt{4(a^2-b)}$ = 2m So , after solving it. I got |$a^2-b|=m^2. $ I’m not able to…
Rider
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Is this what is meant by quadratic equation , function?

It says in my textbook that : Let $f(x) - g(x) = 0$. Then, let this equation be $h(x)$. If $h(x)$ is a quadratic Q.F, then $h(x) = 0$ is a quadratic equation. I need to ask some questions regarding this statement i.e. $1)$ When I say $f(x)-g(x) =…
Rider
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What's the condition required in terms of roots for getting exact three integral negative values for this quadratic?

The number of integral values of a for which $ax^2 - (4-2a)x - 8<0$ for exactly three integral value of x .what i did was find the roots of this equation lets say $x_1$ ,$x_2$ ,will this condition satisfy the criteria: $2<|x_1 -x_2|<4$ and a>0? Or…
user900638
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System of equations in x and y

Solve for $x, y \in \mathbb{R} $ $$ 5x \left(1+\frac{1}{x^2+y^2}\right) =12$$ $$ 5y \left(1-\frac{1}{x^2+y^2}\right) =4$$ I need a Different Approach apart from what i posted..Thank You
Ekaveera Gouribhatla
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