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

5400 questions
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by completing the square find in terms of k the roots of the equation $x^2 + 2kx-7=0$

By completing the square find in terms of $k$ the roots of the equation $$x^2 + 2kx-7=0$$ prove for all real values of $k$, the roots are real
Jess
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sub rectangle region from combined area

I have a rectangle divided like so ---------------------- | | | | | | | | | W | | | | | | ---------------------- L L The total area of both…
Mike McMahon
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Which is the correct method to determine value of expression $\frac{p^{15} + p^{11} + q^{15} + q^{11}}{p^{13} + q^{13}}$

$p, q$ are the roots of equation $x^2 - 5x + 1 = 0$ and we need to calculate the value of expression $E = \frac{p^{15} + p^{11} + q^{15} + q^{11}}{p^{13} + q^{13}}$. I will list 2 methods that produce different values, please tell me which one is…
Vineet Mangal
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Finesse vs. brute force in solving quadratic equations

In Higher Algebra by Hall and Knight, the following "artifice" for solving a certain type of equations is given: Solve: $\sqrt{3x^2-4x+34} - \sqrt{3x^2-4x-11} = 9$ They make use of the fact that $(3x^2-4x+34) - (3x^2-4x-11) = 45$, and utilizing…
ankush981
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Why is performing synthetic division of polynomials acceptable when one may be zero? ${}$

So there is a typical problem in quadratic equations. Consider one as an example: If $x=3 +\sqrt 5$ then find $x^4 -12x^3 +44x^2 -48x +17$. The solution is pretty simple and a similar approach is put forward in another problem in complex numbers.…
Techno Freak
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The least value of $4x^2-4ax +a^2-2a+2$ on $[0,2]$ is $3$. What is the integer part of $a$?

The least value of $4x^2-4ax +a^2-2a+2$ on $[0,2]$ is $3$. What is the integer part of $a$? We know that minimum value of a quadratic is $-\cfrac{b}{2a}$. We will get one condition from here and $-\cfrac{b}{2a}$ should be equal to $3$. But the…
maths lover
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$a$ and $b$ are the roots of quadratic equation $x^2 -2cx-5d=0$ and $c$ and $d$ are the roots of quadratic equation $x^2 -2ax-5b=0 $

Let $a,\,b,\,c,\,d$ be distinct real numbers and $a$ and $b$ are the roots of quadratic equation $x^2 -2cx-5d=0$ and $c$ and $d$ are the roots of quadratic equation $x^2 -2ax-5b=0$. Then find the value of $a+b+c+d$. I could only get $2$ equations…
maths lover
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How to convert a quadratic solution to an unusual format

I'm looking at old past papers and found this question: "Solve the quadratic equation $3x^2 + 4x - 5$ giving your answer in the form $\frac{a}{b\pm\sqrt{19}}$, where $a$ and $b$ are integers." I've never seen a quadratic solution in this form,…
KD97
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How does one find values of $m$ for which the roots of $2x^2-mx-8=0$ differ by $m-1$

Find values of $m$ for which the roots of $2x^2-mx-8$ differ by $m-1$. When I attempted to solve this I tried to simplify it into something like this: $((m-1)+2)((m-1)-4)$ but when I expanded I got $m^2 -4m -5$ which is nowhere near $2x^2-mx-8$ so…
allan
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Proving the quadratic formula (for dummies)

I have looked at this question, and also at this one, but I don't understand how the quadratic formula can change from $ax^2+bx+c=0$ to $x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}$. I am not particularly good at maths, so can someone prove the quadratic…
imulsion
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Given $x^2 - 3x +6=0$ find $x^4$ in terms of $x$.

Could someone please explain what this question is asking? I might be able to solve it by myself by what does it mean by $x^4$? I interpret it as provide a value of $x^4$ given this quadratic so quadratic formula then raise both sides to power $4$ ?…
user71207
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Is a Quadratic equation a function?

The definition of a function is "A function is a relation in which there is never more then one value of the dependent variable for every value of the independent variable." Since a quadratic equation has two solutions for every input does this…
Antonio
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Is it possible to prove $a \in\mathbb R$ in the quadratic?

$a^2−2a+17>0$ I didn't really know how to go about proving this. At first I tried finding the range of '$a$' but the equation has imaginary roots so that doesn't really help. Trying to look at it graphically, it's a parabola that doesn't touch the…
Selfish Stoic
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$y=\dfrac{2x^2+3x-4}{-4x^2+3x+2}$. Find its horizontal asymptotes?

$y=\dfrac{2x^2+3x-4}{-4x^2+3x+2}$. Find its horizontal asymptotes? My attempt is as follows:- Horizontal asymptotes are the horizontal lines which signify the values of $y$ which graph cannot ever attain. There are two ways to find horizontal…
prat
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Why does't quadratic formula work to factor polynomial when $a \ne 1$?

$$2x^2 + 3x + 1$$ applying quadratic formula: $$x = \frac{-b\pm \sqrt{b^2-4ac}}{2a}$$ $$a=2, b=3, c=1$$ $$x = \frac{-3 \pm \sqrt{3^2 - 4 \cdot 2 \cdot1}}{2\cdot2}$$ $$x = \frac{-3 \pm \sqrt{9-8}}{4}$$ $$x = \frac{1}{4}[-3 +…
Bill Moore
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