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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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Set of all real values of $a$ for which the equation $(a-4)\sec^4x+(a-3)\sec^2x+1=0, (a\ne4)$ has real solutions

Find set of all real values of $a$ for which the equation $(a-4)\sec^4x+(a-3)\sec^2x+1=0, (a\ne4)$ has real solutions. Let $\sec^2x=t$. So, the equation becomes $(a-4)t^2+(a-3)t+1=0$. Since $\sec^2x\ge1\implies t\ge1\implies$ both the roots of the…
aarbee
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I'm wrestling with this quadratic and trying to figure out how to factor it:

The quadratic equation is $$a^2-9a+14=0$$ I know that the product of the last terms of the binomial for an equation equals the third term of the polynomial. Also, the sum of the products of those two numbers should be the middle (second) term of the…
Lacy Hayes
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Find $k$ for real solutions of quadratic equation $\sin^4 x + 2(1 + k)\sin^2 x- k - 2 = 0$

I'm stuck on this quadratic problem: $$\sin^4 x + 2(1 + k)\sin^2 x- k - 2 = 0$$ $$k = ?$$ I substituted $t = \sin^2(x)$ this gave me $t^2 + 2(1 + k)t - k - 2 = 0$ and than I figured out that the discriminant is $k^2 + 3k + 3 \ge 0$ which is true for…
Vojtěch Kříbek
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Difficulty with Po-shen Loh Quadratic Equation

I am a math tutor looking for a better way to explain quadratics to my students. I came across the so-called "Po-shen Loh" method on YouTube, and I tried to apply it to a quadratic equation that one of my students was having trouble with. (Here is…
TookieWookie
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Quadratic Equations Mathematical Method

The question is : If $a,b$ are integers such that all the roots of the equation $ ( x^2+ax+20)(x^2+17x+b)=0 $ are negative integers. What are the smallest possible vaues of $a,b$? My approach to this goes like this: Either $( x^2+ax+20)=0$…
user880107
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Quadratic Formula problem?

There is a right triangle. The hypotenuse is 17 units. The sum of the other two sides is 23. Find the length of the two other sides. Thanks for everyone's help in advance!
Rose
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Derivation for finding vertex and turning point of parabola?

My teacher told me that form the equation of parabola . $y + \frac{D}{4a} = a(x+\frac{b}{2a})^2$ So , to find vertex of turning point , Make $y=0$ and $x = 0$. You get $-\frac{b}{2a}$ and $-\frac{D}{4a}$ but aren’t we getting these values in +? I am…
Srijan
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Solving quadratic equation for polynomial coefficients

I am trying to solve an assignment where the zeros of a PID-controller is given and the polynomial controller-constants need to be found. The polynomial form of a standard PID-controller should be defined as: $\dfrac{Kd\cdot s^2+Kp\cdot s +…
Kristian Slot
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Getting back to the quadratic equation again

Indian mathematician Shridharacharya and Persian Mathematician Omar Khayyam (may be others also) are known to have found the formula for roots of a quadratic equation $ax^2+bx+c=0$ as $$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}~~~~(1)$$ independently by…
Z Ahmed
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Let a, b and c be odd positive integers . Show that the quadratic equation ^2++=0 has no rational solution.

To prove this, I think the Δ should =$k^2$ so I let a=2p-1, b=2q-1, c=2r-1, where p, q, r are all positive integers, then I calculated $ b^2-4ac$ which is $-16 p r + 8 p + 4 q^2 - 4 q + 8 r - 3$ and find it hard to prove that $-16 p r + 8 p + 4 q^2…
SaltedFishKing
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Conceptually(not algebraically), why do I get a quadratic equation when I subtract a linear equation form a quadratic equation.

While simultaneously solving a quadratic equation and a linear equation, I noticed that their intercepts are actually equal to the x intercepts of another quadratic function. I find it hard to grasp that subtracting a linear equation from a…
ashir
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If $kx^2-4x+3k+1>0$ for at least one $x>0$ and if $k\in S$, find $S$

Options ; $A)~ (1,\infty)~~ B)~(0,\infty)~~C)~(-1,\infty)~~D)~(-\frac 14 , \infty)$ Obviously $$16-4(k)(3k+1)<0$$ $$k\in (-\infty, -\frac 43)\cup (1,\infty)$$ And also $k>0$ so the answer should be A) The answer is, however, A, B, D I think it has…
Aditya
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Where did 3.5 come from?

This is a homework question. The problem. I know the solution, but I don't know where it came from. The videos say nothing. The equation is $d(v) = (2.15v^2)/(64.4f)$ I need to solve for $f$, so I tried plugging in the numbers from the table into…
user800887
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Application of quadratic functions to measurement and graphing

thanks for any help! Q1. Find the equation of the surface area function of a cylindrical grain silo. The input variable is the radius (r). (the equation is to be graphed using a graphics calculator in the following question) Height (h) = 5…
David
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Integral roots in quadratic equation

The smallest possible natural number $n$, for which the equation $x^{2}-n x+2014=0$ has integral roots, is I know the discriminant will be a perfect square, but I am struck on equation of discriminant.
Piesquareisg
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