# Finding $b(a+c)$ for Real Roots of $\sqrt{2014}x^3-4029x^2+2=0$

• MHB
• anemone
In summary, to find the value of $b(a+c)$ in the given equation, use the quadratic formula with the values of $a=\sqrt{2014}$, $b=-4029$, and $c=2$. There are various methods, such as factoring, completing the square, and using the quadratic formula, that can be used to solve this type of equation. To check if the solution for $b(a+c)$ is correct, plug the value back into the original equation. The equation can only have one solution for $b(a+c)$ and there are restrictions on the values of $a$, $b$, and $c$ in order for the equation to have real roots.
anemone
Gold Member
MHB
POTW Director
Let $a>b>c$ be the real roots of the equation $\sqrt{2014}x^3-4029x^2+2=0$. Find $b(a+c)$.

To avoid radicals let $\sqrt{2014}=p$
So we get $px^3-(2p^2+1)x^2 +2 = 0$
Or factoring we get $(px-1)(x^2-2px-2)$ = 0
So one root is $x= \frac{1}{p}$ and other two roots are roots of $x^2-2px-2=0$
For the equation $x^2-2px-2=0$ sum of the roots is 2p and product is -2. so one root has to be -ve and
the postiive root shall be above 2p
So $b=\frac{1}{p}\cdots(1)$
And c is the -ve root and $a> 2p$
a,c are roots of $x^2-2px-2=0$ so $a+c = 2p\cdots(2)$
Hence $b(a+c) = 2$ using (1) and (2)

## 1. What is the formula for finding $b(a+c)$?

The formula for finding $b(a+c)$ in the equation $\sqrt{2014}x^3-4029x^2+2=0$ is $b(a+c) = \frac{-b}{a}$, where $a$ and $b$ are the coefficients of the quadratic term and the linear term, respectively.

## 2. How do you know if a given equation has real roots?

A given equation has real roots if the discriminant, $b^2-4ac$, is greater than or equal to 0. If the discriminant is less than 0, the equation will have complex roots.

## 3. Can the value of $b(a+c)$ be negative?

Yes, the value of $b(a+c)$ can be negative. This would occur if the coefficient of the linear term, $b$, is negative and the coefficient of the quadratic term, $a$, is positive.

## 4. What is the significance of finding $b(a+c)$ for real roots?

Finding $b(a+c)$ for real roots is significant because it allows us to determine the sum of the roots of the equation. This can help us in solving the equation and understanding the behavior of the graph of the equation.

## 5. Can the value of $b(a+c)$ be 0?

Yes, the value of $b(a+c)$ can be 0. This would occur if the coefficient of the linear term, $b$, is 0. In this case, the equation would simplify to $ax^2+c=0$ and the value of $b(a+c)$ would be 0 regardless of the value of $a$ and $c$.

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