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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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Quadratic equations / Perfect square

I am dealing with a programming problem and was wondering if there's a general method to solve it. Is there a general way to find 'n' such that: $a^2n^2 + bn - c$ is a perfect square? a, b, c given numbers. In my problem the first term is always a…
Latin PMB
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Proof for showing $1 + \frac{c}{a} + \frac{b}{a} > 0$ for any quadratic equation,where its roots are non real.

The question is as follows : If the equation $ax^2 + bx + c$ has non real roots, prove that $1 + c/a + b/a > 0 $. Looking at the question,the first thing that came to my mind was to use the relationship that the discriminant is less than 0…
Aditya Pai
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Interval of $a$ for which the solutions of the equation $x^2-6ax+2-2a+9a^2$ are bigger than $3$

The question is asked so that I have multiple choices and need to prove the thing both ways (it's an equivalence). The problem is, whichever thing I compare to the minimal values of the roots I try to use, whether it be Viete's formula, the…
John Doe
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Find the equation whose roots are each six more than the roots of $x^2 + 8x - 1 = 0$

Find the equation whose roots are each six more than the roots of $x^2 + 8x - 1 = 0$ I must use Vieta's formulas in my solution since that is the lesson we are covering with our teacher. My solution: Let p and q be the roots of the…
user164403
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Solving an operation involving roots of a quadratic equation

A question from my book: $3x^2 + 7x + 5 = 0$ So, $\sqrt{(x_1^2 + 2x_1x_2 + x_2^2)} + x_1x_2 = ?$ Options: A) $4$ B) $5$ C) $6$ D) $7$ E) $8$ It's looking too easy, my answer is $-\frac{2}3$, but it does not exist in options and my book says that…
Oğuz İsmayil uysal
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Finding the positive component of a constant in a quadratic equation.

Can you help me to do this question:it is from a past cambridge exam paper Find the positive constants $a$ and $b$ such that $x^4+9/x^4 =[x^2-a/x^2 ]^2+b$ for all non-zero values of $x$. Hence write down, or obtain otherwise, the least possible…
Sophia Granger
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Does the quadratic formula always work for a quadratic?

Does the quadratic formula always work for a quadratic? If it does, why are the factors sometimes imaginary numbers?
Alex
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How to find the equation for a parabola for which you are given two points and the vertex?

I was originally given the value $(4,-2)$ as the vertex of a parabola and told that it also includes the value $(3,-5)$. From this point, I deduced that the next point would have the same y-value as the point whose x-value is equidistant from the…
Yasmine
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How do you find out the range of values when dealing with simultaneous equations?

Find the range of value for $k$ for which $kx + y = 3$ meets $x^2 + y^2 = 5$ in two distinct points. im so stuck can someone give me a clear guide to the correct method and answer, thank you
Kelly
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quadratic equation

A garden is in the shape of a rectangle, $20$m by $8$m. Around the outside is a border of uniform width and in the middle is a square pond. The area which is not occupied by either border or pond is 1$24 m^2$. Letting the width of border be $x$cm,…
mathlover
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Find the range of $k$ in $f(x) = \frac{x^2-k}{x-2}$

I have the following question: For real $x$, $f(x) = \frac{x^2-k}{x-2}$ can take any real value. Find the range of values $k$ can take. Here is how I commenced: $$ y(x-2) = x^2-k \\ -x^2 + xy - 2y + k = 0\\ $$ So we have $a=-1$, $b=y$,…
hohner
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Deriving coefficient q in reduced quadratic equation using Vieta's formulas

I have the following example of the use of Vieta's formulas in my textbook: Let's have a quadratic equation of the form $x^2+x+q=0$ The following conditions apply: the equation has two roots, such as $x_1^2+x_2^2=1$. We need to find $q$. The…
CopperKettle
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Quadratic Formula: Word Problem.

A person decides to build a horse corral using a barn for one side. Has has 30m of fencing materials and wants the corral to have an area of 100m^2. What are the dimensions of the corral? Let width = x Let length = y y=30-2x. That's all I could…
Dev
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Quadratic Functions Word Problem

A holding pen is being built alongside a long building. The pen requires only three fenced sides, with the building forming the fourth side. There is enough material for 90m of fencing. Predict what dimensions will give the maximum area of the pen…
Dev
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For the given quadratic equation find the value of p

For the equation 3x^2 + px + 3 = 0 , p>0, if one of the roots is square of the other, then p is equal to? Solving the equation, i get the value of p as -6 but the question states that p>0. Is there another way of solving this or is the question…
Mohit Singh
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