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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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Solve for x: $x^2+2bx-a^2+8ab-15b^2=0$

$$x^2+2bx-a^2+8ab-15b^2=0$$ I am having a problem with solving these kinds of equations. I just get confused and I don't see what I need to do. Can somebody help, give some trick how to solve these equations easily?
RiktasMath
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Quadratic equation with different indices

My maths teacher gave me a worksheet to work through as I was getting slightly bored in lessons. However, there was one question which I cannot do. The worksheet gives the answer, but you are supposed to show how you did it. Here is the question: …
AJ123
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Systems of equations algebraically using the quadratic formula

$$y=-x^2+2x+9$$ $$y=-5x^2+10x+12$$ Round answer to two decimal places. So far I made both equations equal the other which lead to $-4x^2+8x+12$, took $4$ out, $4 (-x^2+2x+3)$. Then that bracketed terms were put into the quadratic formula to equal…
Grimestock
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Find all real solutions of $\sqrt{x} - \sqrt{2-2x} = 1$

Squaring both sides of $\sqrt{x} - \sqrt{2-2x} = 1$ and rearranging I arrive at the quadratic $9x^2 - 10x + 1 = 0$ which has solutions $x=1/9$ and $x=1$. I don't understand why $x=1$ fits the original equation but $x=1/9$ doesn't (left hand side…
user7875067
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When a polynomial is ÷ by another polynomial and it gives a remainder but when we put the value of the x in the polynomials it gives 0 remainder.

Let we have an equation $x^2+4x+2$ and we want to divide it with $x-4$. By remainder theorem it gives the value of $34$. But when we put the value of $x=6$ in equations it gives zero remainder. And quotient is $31$. How's that? Where I am mistaking?
Bilal Sheikh
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How to find the increasing and decreasing intervals of a quadratic equation without calculus

I want to find the increasing and decreasing intervals of a quadratic equation algebraically without calculus. The truth is I'm teaching a middle school student and I don't want to use the drawing of the graph to solve this question.
user42912
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can u put this equation in function of $H$ : $T = \sqrt{h_{1}^2 - H^2}+\sqrt{h_{2}^2 -H^2}$

can u put this equation in function of H : $$T = \sqrt{h_{1}^2 - H^2}+\sqrt{h_{2}^2 -H^2}$$ to: H = something... $T$ and $H$ are variables, and $h_1$ and $h_2$ are constants thank u all :D
Miguel Silvestre
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Solving for a variable in an expression

I have got this expression $X = 20t + 0.5t^2$ and the requirement for this is to solve for t. I have tried to do this by factoring t out but was not able to do it. This is what i have done: $X = 20t + 0.5t^2\\$ $X = (0.5t +20)t $
Kabit
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Find all real values of 'a' for which all the function roots are integers.

Find all real values of 'a' for which all the function roots are integers. $$ f(x) = ax^{2} + (a+1)x + a-1$$ I was thinking about Vieta's formula so: $$ \\xy = 1 - \frac{1}{a} \\x + y = -1 - \frac{1}{a} \\xy ∈ \Bbb Z \\x+y ∈ \Bbb Z. \\a ∈ \Bbb…
VereX
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Solving System of quadratic equations

If $b²-4ac=0$ ($a \neq 0$, and $a, b, c \in \mathbb {R}$) and $x, y $ satisfy the system $$ax²+(b+3)x+c=3y$$ and $$ay²+(b+3)y+c=3x$$ then the value of $x/y$ is...?
Rohan Shinde
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find pairs of real numbers $x, y$ to satisfy this equation

equation is: $(x + y)^2 = (x + 3)(y − 3)$ I'm not asking for a solution, but an approach. How do I prove this kind of question? I have tried to arrange it so that it is $x + y$ = .... But I still get nothing, nothing intuitive at least. What is…
Po Chen Liu
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Find the set of values of $a$ for which the inequality $(x-3a)(x-3-a)<0$ is satisfied for all $x$ in the interval $1\le x\le3$.

Find the set of values of $a$ for which the inequality $(x-3a)(x-3-a)<0$ is satisfied for all $x$ in the interval $1\le x\le3$. $$(x-3a)(x-3-a)<0\implies a+31$ and $3a<3$ ,so $a\in (-2,1)$ but the…
learner_avid
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Expressing variable q in terms of p. Where p and q are contents in a quadratic.

Suppose that $p$ and $q$ are constants such that the smallest possible value of $x^2+px+q$ is $0$. Express $q$ in terms of $p$. I am unsure what it is asking. I feel it is asking something very simple. However I am unsure of what it is. Any…
user3753
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Value of this expression

If $\alpha$ and $\beta$ are the roots of the equation $$x^2 + x − 3 = 0$$ find the value of the expression $4\beta^2 − \alpha^3$. I tried using sum of roots and product of roots formulas but could not get the answer.
KBC
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Implications of the two solutions in a quadratic

My question is derivative of this one here that caused me to get an entirely new question that came out of my working on it and reading some very helpful answers. Two quadratic equations have real roots $\alpha$ and $\beta$ such that $$\alpha -…
sangstar
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