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 requires applying the quadratic formula or factoring techniques to find the values of x that satisfy the equation.

Answer

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

Steps to Solve

Identify coefficients in the quadratic formula
For the quadratic equation $x² - 3x - 3 = 0$, identify the coefficients:

$a = 1$ (coefficient of $x²$)
$b = -3$ (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 in the identified coefficients:
$$ x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4 \cdot 1 \cdot (-3)}}{2 \cdot 1} $$
Calculate the discriminant
First, calculate the discriminant $b^2 - 4ac$:
$$ (-3)^2 - 4(1)(-3) = 9 + 12 = 21 $$
Substitute the discriminant back in
Now substitute the discriminant back into the quadratic formula:
$$ x = \frac{3 \pm \sqrt{21}}{2} $$
Write the final solutions
This gives two possible solutions:
$$ x = \frac{3 + \sqrt{21}}{2} \quad \text{and} \quad x = \frac{3 - \sqrt{21}}{2} $$

The solutions are
$$ x = \frac{3 + \sqrt{21}}{2} \quad \text{and} \quad x = \frac{3 - \sqrt{21}}{2} $$

More Information

The quadratic formula can always be used to find the roots of any quadratic equation, regardless of whether it can be factored easily. The solutions found here are also exact values in terms of square roots.

Tips

Misidentifying coefficients: Ensure $a$, $b$, and $c$ are correctly identified from the equation.
Incorrectly calculating the discriminant: Make sure to square the coefficients and multiply properly.
Forgetting to divide by $2a$: Always ensure to divide by $2$ when applying the formula.

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