Finding Difference Between Roots of $x^2 - 6x - 16 = 0$

To find the difference between the roots of the quadratic equation $x^2 - 6x - 16 = 0$, we can utilize a formula derived from the properties of quadratic equations.

Identify Quadratic Equation Coefficients

The given equation is $x^2 - 6x - 16 = 0$. Comparing this to the standard quadratic form $ax^2 + bx + c = 0$, we identify the coefficients:

• $a = 1$
• $b = -6$
• $c = -16$

Calculate Discriminant

The discriminant ($D$) is calculated using the formula $D = b^2 - 4ac$. Substituting the coefficient values:

$D = (-6)^2 - 4(1)(-16)$
$D = 36 + 64$
$D = 100$

Difference Between Roots Calculation

The absolute difference between the roots ($\alpha$ and $\beta$) of a quadratic equation $ax^2 + bx + c = 0$ is given by the formula:

$|\alpha - \beta| = \frac{\sqrt{D}}{|a|} = \frac{\sqrt{b^2 - 4ac}}{|a|}

Now, substitute the values of $D$ and $a$ into the formula:

$|\alpha - \beta| = \frac{\sqrt{100}}{|1|}$
$|\alpha - \beta| = \frac{10}{1}$
$|\alpha - \beta| = 10$

The difference between the roots of the equation $x^2 - 6x - 16 = 0$ is 10.