Solve for x: x² - 3x - 3 = 0

Understand the Problem

The question is asking to solve the quadratic equation x² - 3x - 3 = 0 for the variable x. This involves applying the quadratic formula or factoring methods to find the values of x that satisfy the equation.

Answer

The solutions are $x = \frac{3 + \sqrt{21}}{2}$ or $x = \frac{3 - \sqrt{21}}{2}$.

Steps to Solve

Identify coefficients from the equation

For the quadratic equation in the form $ax^2 + bx + c = 0$, identify the coefficients:

$a = 1$ (coefficient of $x^2$)
$b = -3$ (coefficient of $x$)
$c = -3$ (constant term)

Apply the quadratic formula

The quadratic formula is given by:

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

Substituting the identified values:

$$ x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4 \cdot 1 \cdot (-3)}}{2 \cdot 1} $$

Calculate the discriminant

Calculate the discriminant $b^2 - 4ac$:

$$ (-3)^2 - 4 \cdot 1 \cdot (-3) = 9 + 12 = 21 $$

Substitute the discriminant back into the formula

Now substitute this value back into the quadratic formula:

$$ x = \frac{3 \pm \sqrt{21}}{2} $$

Write the final solutions

The solutions for $x$ are:

$$ x = \frac{3 + \sqrt{21}}{2} \text{ and } x = \frac{3 - \sqrt{21}}{2} $$

The solutions for the quadratic equation $x^2 - 3x - 3 = 0$ are:

$$ x = \frac{3 + \sqrt{21}}{2} \text{ or } x = \frac{3 - \sqrt{21}}{2} $$

More Information

These solutions are derived from applying the quadratic formula to the given equation. The solutions can be expressed in decimal form for approximate numerical values, but the exact values are more precise for further calculations.

Tips

Forgetting to check the discriminant: Some may forget to calculate $b^2 - 4ac$, which determines the nature of the roots.
Incorrectly simplifying terms: When substituting back into the quadratic formula, it’s easy to make arithmetic mistakes.
Using wrong values for a, b, or c: Wrongly identifying coefficients can lead to incorrect solutions.

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