If the roots of the equation x² - bx + c = 0 be two consecutive integers, then b² - 4c equals (A) -2

Q: Wait, why do we use the absolute difference $|\alpha - \beta|$ instead of just $\alpha - \beta$?

Great question! The roots $\alpha$ and $\beta$ are interchangeable labels. We don't know which one is larger. By using the absolute difference, we ensure the distance is always positive, which is physically correct for a distance on the number line. It keeps our math robust regardless of which root we call $\alpha$.

Q: Can I just assume the roots are $n$ and $n+1$ and solve it that way?

Absolutely! That is a fantastic, intuitive approach. If the roots are $n$ and $n+1$, then the sum of roots is $2n+1 = b$ and the product is $n(n+1) = c$. If you substitute these into $b^2 - 4c$, you will get $(2n+1)^2 - 4(n^2+n) = 4n^2 + 4n + 1 - 4n^2 - 4n = 1$. Both methods lead to the same beautiful result!

Q: Why does the formula $|\alpha - \beta| = \frac{\sqrt{D}}{|a|}$ work?

It comes from the quadratic formula! The roots are $\frac{-b \pm \sqrt{D}}{2a}$. If you subtract $\frac{-b - \sqrt{D}}{2a}$ from $\frac{-b + \sqrt{D}}{2a}$, the $-b$ terms cancel out, leaving you with $\frac{2\sqrt{D}}{2a}$, which simplifies to $\frac{\sqrt{D}}{a}$. We take the absolute value to handle the sign of $a$.

Question

If the roots of the equation $x^2 - bx + c = 0$ be two consecutive integers, then $b^2 - 4c$ equals

Options:

(A) $-2$

(B) $3$

(C) $2$

(D) $1$

Sigma Percentile · Accepted Answer

### The Beauty of Consecutive Roots

Welcome, future engineer! Today, we are going to peel back the layers of a seemingly simple quadratic equation. It is not just about finding $x$; it is about understanding the geometric soul of the equation $x^2 - bx + c = 0$.

Imagine a parabola dancing on the coordinate plane. The roots are the points where this curve kisses the x-axis. When we say the roots are consecutive integers, we are imposing a rigid, beautiful constraint on the geometry of this parabola.

### The Geometric Constraint

Think about the number line. If your roots are consecutive, they are neighbors. If one is at $2$, the other is at $3$. If one is at $-5$, the other is at $-4$.

In every single case, the distance between them is exactly $1$. Mathematically, if we call the roots $\alpha$ and $\beta$, we can write this as $|\alpha - \beta| = 1$.

This is our anchor. This is the physical reality of the problem.

### The Discriminant Connection

Now, we need to bridge the gap between this geometric distance and the algebraic coefficients $b$ and $c$. We have a powerful tool in our arsenal: the difference of roots formula.

For any quadratic $ax^2 + bx + c = 0$, the distance between the roots is given by:

$$|\alpha - \beta| = \frac{\sqrt{D}}{|a|}$$

Here, $D = b^2 - 4ac$ is the discriminant. This formula is a shortcut through the forest of algebra, connecting the roots directly to the discriminant.

### The Final Synthesis

Let's apply this to our specific equation $x^2 - bx + c = 0$. Here, the leading coefficient $a$ is $1$. The discriminant $D$ is $b^2 - 4(1)(c) = b^2 - 4c$.

Now, look at the magic. We know $|\alpha - \beta| = 1$. We also know:

$$|\alpha - \beta| = \frac{\sqrt{b^2 - 4c}}{1}$$

Equating these, we get $1 = \sqrt{b^2 - 4c}$. To isolate our target expression, we square both sides:

$$1^2 = b^2 - 4c$$

Thus, **$b^2 - 4c = 1$**.

It is elegant, it is simple, and it is profound. Whenever you see a monic quadratic with consecutive integer roots, you now know the discriminant is always $1$. Keep this intuition, and you will conquer any problem that comes your way!

Animated Solution for Mathematics - Quadratic Equations: If the roots of the equation $x^{2} - b x + c = 0$ be two consecutive integers, then $b^{2} - 4 c$ equals

Select Answer:

Visualized Solution

The Quadratic Equation

Consecutive Integer Roots

Mathematical Translation of Consecutive

The Difference of Roots Formula

Identifying the Coefficients

Calculating the Discriminant

Setting up the Equation

Solving for the Target Expression

Final Conclusion

The Sigma Insight: Relation Between Roots and Coefficients

The Beauty of Consecutive Roots

The Geometric Constraint

The Discriminant Connection

The Final Synthesis

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Animated Solution for Mathematics - Quadratic Equations: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the roots of the equation x2−bx+c=0 be two consecutive integers, then b2−4c equals