Introduction

To find the constant k for a quadratic equation such that one root exceeds the other by 5, we need to use the properties of quadratic equations and algebraic manipulations. This article explores the process step-by-step and provides a detailed explanation and conclusion.

Solving the Equation: x^2 - x - 2k 0

Consider the quadratic equation x^2 - x - 2k 0. Let's denote the roots of this equation as r_1 and r_2. According to the problem, we know that one root exceeds the other by 5.

Step 1: Express the Roots

Let's denote the roots as r_1 r_2 5. Using Vieta's formulas, we know that the sum and product of the roots can be expressed in terms of the coefficients of the equation.

Step 2: Apply Vieta's Formulas

From Vieta's formulas, we have:

Substituting r_1 r_2 5 into the sum of the roots formula gives us:

(r_2 5) r_2 1

This simplifies to:

2r_2 5 1

Subtracting 5 from both sides:

2r_2 -4

Dividing by 2:

r_2 -2

Now, substituting r_2 -2 into the equation for r_1.

r_1 r_2 5 -2 5 3

So, the roots are r_1 3 and r_2 -2.

Step 3: Find the Constant k

Using the product of the roots, we have:

r_1 cdot r_2 -2k

Substituting r_1 3 and r_2 -2:

3 cdot -2 -6

Since -6 -2k, solving for k:

-2k -6

Dividing both sides by -2:

k 3

Therefore, the constant k is

Verification and Additional Insights

To verify, we can factor the quadratic equation x^2 - x - 6 0. The roots are:

x 3 and x -2

The difference between the roots is:

3 - (-2) 5

This confirms our solution. Additionally, using the quadratic formula,

x frac{1 pm sqrt{1 8k}}{2}

Setting k 3 gives us:

x frac{1 pm sqrt{25}}{2} frac{1 pm 5}{2}

This also confirms that the roots are x 3 and x -2.

Conclusion

The constant k for the quadratic equation x^2 - x - 2k 0, such that one root exceeds the other by 5, is