solve the general quadratic equation for a b and c where x = +- root 3

To solve the general quadratic equation where \(x = \pm \sqrt{3}\), we can start by using the fact that \(x = \pm \sqrt{3}\) are the roots of the quadratic equation. Given that the roots of a quadratic equation \(ax^2 + bx + c = 0\) are \(r_1\) and \(r_2\), we can use the **factored form** of the quadratic equation:

\[
a(x - r_1)(x - r_2) = 0
\]

Here, the roots \(r_1 = \sqrt{3}\) and \(r_2 = -\sqrt{3}\) lead to:

\[
a(x - \sqrt{3})(x + \sqrt{3}) = 0
\]

Now, expand the factored form:

\[
a\left( (x - \sqrt{3})(x + \sqrt{3}) \right) = 0
\]

Using the difference of squares identity:

\[
a(x^2 - (\sqrt{3})^2) = 0
\]

This simplifies to:

\[
a(x^2 - 3) = 0
\]

Finally, expanding gives the general quadratic equation:

\[
ax^2 - 3a = 0
\]

Comparing this to the standard form of a quadratic equation \(ax^2 + bx + c = 0\), we can identify:

- \(a = a\)
- \(b = 0\) (since there is no \(x\)-term)
- \(c = -3a\)

Thus, the general form of the quadratic equation where \(x = \pm \sqrt{3}\) is:

\[
ax^2 - 3a = 0
\]

If you have any questions or would like more details, feel free to ask!

Here are 5 related questions to expand on this:

1. How do you derive the sum and product of the roots from a quadratic equation?
2. What happens when the discriminant of a quadratic equation is zero?
3. How can you determine the nature of the roots of a quadratic equation?
4. How would the equation change if the roots were \(x = \pm 2\) instead of \(x = \pm \sqrt{3}\)?
5. How can you graph the quadratic equation \(x^2 - 3 = 0\)?

**Tip:** The discriminant \(b^2 - 4ac\) of a quadratic equation determines the type of roots: real and distinct, real and equal, or complex.

solve the general quadratic equation for a, b, and c where x = ±√3

Discover how to solve the general quadratic equation where the roots are x = ±√3. This step-by-step solution uses key algebraic concepts like the difference of squares and factored forms. Perfect for students in Grades 9-10 looking to deepen their understanding of quadratic equations.

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