Finding the Value of q in a Quadratic Equation

The problem asks us to find the value of 'q' in a quadratic equation $x^2 - 8x + q = 0$, given that its roots are $\alpha$ and $\beta$, with the condition $\alpha > \beta$, and that $\alpha^2 - \beta^2 = 16$.

Understanding Quadratic Equation Roots

For a general quadratic equation in the form $ax^2 + bx + c = 0$, the sum and product of its roots (let's call them $\alpha$ and $\beta$) are related to the coefficients as follows:

• Sum of roots: $\alpha + \beta = -b/a$
• Product of roots: $\alpha \beta = c/a$

Applying Properties to the Given Equation

The given equation is $x^2 - 8x + q = 0$. Comparing this to the standard form $ax^2 + bx + c = 0$, we have $a=1$, $b=-8$, and $c=q$.

Using the root properties:

• Sum of roots: $\alpha + \beta = -(-8)/1 = 8$
• Product of roots: $\alpha \beta = q/1 = q$

So, we know that $\alpha + \beta = 8$ and $\alpha \beta = q$. Our goal is to find the value of q, which means we need to find the product of the roots.

Using the Given Condition $\alpha^2 - \beta^2 = 16$

We are given the condition $\alpha^2 - \beta^2 = 16$. This expression is a difference of squares, which can be factored as $(\alpha - \beta)(\alpha + \beta)$.

So, we can rewrite the condition as:

$(\alpha - \beta)(\alpha + \beta) = 16$

Solving for $\alpha$ and $\beta$

We know from the sum of roots property that $\alpha + \beta = 8$. We can substitute this value into the factored condition:

$(\alpha - \beta)(8) = 16$

Now, we can solve for the difference between the roots, $\alpha - \beta$:

$\alpha - \beta = 16 / 8$

$\alpha - \beta = 2$

Now we have a system of two linear equations with two variables, $\alpha$ and $\beta$:

1. $\alpha + \beta = 8$
2. $\alpha - \beta = 2$

We can solve this system by adding the two equations:

$(\alpha + \beta) + (\alpha - \beta) = 8 + 2$

$2\alpha = 10$

$\alpha = 10 / 2$

$\alpha = 5$

Now substitute the value of $\alpha$ back into the first equation ($\alpha + \beta = 8$):

$5 + \beta = 8$

$\beta = 8 - 5$

$\beta = 3$

We have found the roots to be $\alpha = 5$ and $\beta = 3$. The condition $\alpha > \beta$ (5 > 3) is satisfied.

Finding the Value of q

We know that $q$ is the product of the roots, $\alpha \beta$. Now that we have found $\alpha = 5$ and $\beta = 3$, we can calculate q:

$q = \alpha \beta$

$q = (5)(3)$

$q = 15$

Therefore, the value of q is 15.

Revision Table: Key Steps to Solve Quadratic Root Problems

Additional Information: Discriminant of a Quadratic Equation

While not directly used in this specific problem, the discriminant of a quadratic equation ($ax^2 + bx + c = 0$) is $\Delta = b^2 - 4ac$. The discriminant tells us about the nature of the roots:

• If $\Delta > 0$, the equation has two distinct real roots ($\alpha \ne \beta$).
• If $\Delta = 0$, the equation has exactly one real root (or two equal real roots) ($\alpha = \beta$).
• If $\Delta < 0$, the equation has two complex roots (non-real roots).

In our problem, the roots are distinct ($\alpha > \beta$), which implies the discriminant must be positive. Let's check: $b^2 - 4ac = (-8)^2 - 4(1)(q) = 64 - 4q$. For distinct real roots, $64 - 4q > 0 \implies 64 > 4q \implies q < 16$. Our calculated value $q=15$ is less than 16, which is consistent with distinct real roots