Expression involving roots of quadratic equation

In summary, the expression $\frac{1}{\alpha^2}+\frac{1}{\beta^2}$ can be evaluated by using the sum and product of roots formula for a quadratic equation. The result is equivalent to $\frac{b^2-2ac}{c^2}$.
  • #1
mathdad
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Given that $\alpha$ and $\beta$ are the roots of a quadratic equation, evaluate $\frac{1}{\alpha^2}+\frac{1}{\beta^2}$.

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I've edited your post to include the problem statement in the body of the post and to give it a meaningful title. Please try to do so on your own in the future. :)

RTCNTC said:
Given that $\alpha$ and $\beta$ are the roots of a quadratic equation, evaluate $\frac{1}{\alpha^2}+\frac{1}{\beta^2}$.
$$\gamma(x-\alpha)(x-\beta)=\gamma x^2-\gamma(\alpha+\beta)x+\gamma\alpha\beta$$

Using $ax^2+bx+c$, the sum of the roots is $-\frac{b}{a}$ and the product of the roots is $\frac{c}{a}$.

The given expression with a common denominator is $\frac{\alpha^2+\beta^2}{\alpha^2\beta^2}$.

That expression evaluates to $\frac{b^2-2ac}{c^2}$.
 

FAQ: Expression involving roots of quadratic equation

1. How can I simplify an expression involving roots of quadratic equations?

To simplify an expression involving roots of quadratic equations, you can use the quadratic formula or factor the equation to find the roots. Then, you can substitute the roots into the expression to simplify it.

2. Can the roots of a quadratic equation be imaginary?

Yes, the roots of a quadratic equation can be imaginary if the discriminant (b^2 - 4ac) is negative. In this case, the quadratic equation will have two complex roots, which can still be used to simplify the expression.

3. How do I know if an expression involving roots of quadratic equations is equivalent to a simpler form?

To determine if an expression involving roots of quadratic equations is equivalent to a simpler form, you can substitute the roots into the expression and simplify. If the resulting expression is the same as the original, then the expression is equivalent to a simpler form.

4. Can I solve for the roots of a quadratic equation using only one root?

Yes, you can find the other root of a quadratic equation if you are given one root. To find the other root, you can use the fact that the product of the roots of a quadratic equation is equal to the constant term divided by the coefficient of the squared term.

5. What is the difference between the discriminant and the roots of a quadratic equation?

The discriminant of a quadratic equation is the value inside the square root of the quadratic formula (b^2 - 4ac). It is used to determine the nature of the roots (real, equal, or complex) and can also be used to find the roots. The roots of a quadratic equation are the values that satisfy the equation when substituted into it.

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