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

5400 questions
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Universal quadratic formula?

Is there any way to write the quadratic formula such that it works for $ac= 0$ without having to make it piecewise? The traditional solution of $x = (-b \pm \sqrt{b^2 - 4ac}) / 2a$ breaks when $a = 0$, and the less-traditional solution of $x = 2c /…
user541686
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how to calculate roots of given equation below?

Without solving equation $2x^2 + 9x + 9 = 0$, show that one of the root of the equation is twice the other.
Murtuza Vadharia
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Derivation of the quadratic equation

So everyone knows that when $ax^2+bx+c=0$,$$x = \dfrac{-b\pm\sqrt{b^2-4ac}}{2a}.$$ But why does it equal this? I learned this in maths not 2 weeks ago and it makes no sense to me
user2751288
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Find all values of a for which the equation $x^4 +(a-1)x^3 +x^2 +(a-1)x+1=0$ possesses at least two distinct negative roots

Find all values of a for which the equation $$x^4 +(a-1)x^3 +x^2 +(a-1)x+1=0 $$ possesses at least two distinct negative roots. I am able to prove that all roots would be negative .How to proceed after this.
maths lover
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How do you solve $4x^2=-16x$? I get different answers depending on the method used.

I'm solving the following GRE problem: Solve $4x^2=-16x$ Method 1: I simply divide both sides by $4x$ :$$x=-4$$ Method 2: I solve by factoring:$$4x^2+16x=0$$ $$4x(x+4)=0$$ $$x=-4, x=0$$ Using method 1, I did not get $x=0$ as a solution. Is method 1…
Joebevo
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Prove that $|a|+|b|+|c|\le17$ if $p(x)=ax^2+bx+c$ is a real polynomial with $|p|\le1$ for $0\le x\le1$

Let $ax^2+bx+c$ be a quadratic polynomial with real coefficients such that $$|ax^2+bx+c| \leq 1,$$ for $ 0\leq x\leq 1$. Prove that $$|a|+|b|+|c|\leq 17$$ How to proceed in this particular question. Sorry I can't show any work because I really not…
H.P. Das
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location of roots of quadratic with natural coefficients

the quadratic equation $ ax^2-bx+c=0 $ ; $a,b,c \in \mathbb{N}$, has two distinct real roots belonging to the interval $(1,2)$ , then what would be least value of $a$ and $b$? I tried to solve these four inequalities with iterations,which seems to…
vikas meena
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Discriminant of quadratic function

Prove that if we have two quadratic function $f(x)$ and $g(x)$ such that $|f(x)| >|g(x)|$, then $|\Delta_f|>|\Delta_g| $. I'm looking for hints, I've no idea where should I start. Thanks in advance...
user263286
6
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Quadratic formula - check my simplificaiton

I am trying to solve this equation using the quadratic formula: $$x^2 + 4x -1 = 0$$ I start by substituting the values into the quadratic formula: $$x = {-(4) \pm \sqrt {(4)^2 - 4(1)(-1)} \over 2}$$ which becomes $$x = {-4 \pm \sqrt{20} \over…
dagda1
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quadratic equation what am I doing wrong?

solve $$ \sqrt{5x+19} = \sqrt{x+7} + 2\sqrt{x-5} $$ $$ \sqrt{5x+19} = \sqrt{x+7} + 2\sqrt{x-5} \Rightarrow $$ $$ 5x+19 = (x+7) + 4\sqrt{x-5}\sqrt{x+7} + (x+5) \Rightarrow $$ $$ 3x + 17 = 4\sqrt{x-5}\sqrt{x+7} \Rightarrow $$ $$ 9x^2 + 102x + 289 =…
Gravity
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Where did $-4x$ come from?

I'm going over my quadratic equations for the ACT and I came across this quadratic: $$(x – 2)^2 – 12$$ My teacher said we could have factored it out into this: $$x^2 – 4x – 8$$ But I just don't understand where he got the $-4x$! Help?
Charles Shick
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Find the value of a + b + c + d

Let $a$ and $b$ be the roots of the equation: $x^2 - 10cx - 11d = 0$ where $c$ and $d$ be the roots of $x^2 - 10ax - 11b = 0$. Find the value of $a+b+c+d$, assuming that they all are distinct. I first tried making an equation with roots $(a+b)$ and…
Harshal Gajjar
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Proving the second root of a quadratic equation

If $\alpha$ is a root of the equation $4x^2+2x-1=0$, then prove that $4\alpha^3-3\alpha$ is the other root. How do I proceed? The sum of the roots, the product of the roots lead me nowhere. Should I find the roots of the equation and substitute in…
Tejas
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A quadratic function $f(x)$ satisfies the inequality $-1 < f(x) < 1$ for $x \in [0, 1]$. What can we say about the range of its coefficients?

Let a function $f(x) = ax^2 + bx + c$, where $a, b, c \in R$, satisfy $-1 \leq f(x) \leq 1$ for all $x \in [0, 1]$ then which of the following conclusions can be made? A) $|a| \leq 8$ B) $|b| \leq 8$ C) $|c| \leq 1$ D) $|a| + |b| + |c| \leq 17$ I…
Kremlin
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If $a+b=x$ and $ab=y$, what is the quickest way to solve for $a$ and $b$?

The mechanistic approach would be to simply substitute $b=y/a$ in the first equation to obtain a quadratic in $a$. But seeing the simplicity of the givens, I feel that there must be some better and elegant ways to do this. The best way I could think…
Alraxite
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