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Animated Solution for Mathematics - Quadratic Equations: If the difference between the roots of the equation $x^{2} + a x + 1 = 0$ is less than $5$ , then the set of possible values of $a$ is

Q: Why do we square the inequality $|\alpha - \beta| < \sqrt{5}$? Can't we just work with the modulus directly?

Working with the modulus is certainly possible, but it often leads to complex case-based analysis. Squaring both sides is a powerful, elegant technique here because both sides are positive, which ensures the inequality sign remains unchanged. It effectively removes the square root and the modulus, simplifying the expression into a straightforward quadratic inequality.

Q: Why is the final interval $(-3, 3)$ open? Shouldn't it be closed?

That is a great observation! The problem specifies that the difference between the roots is *strictly* less than $\sqrt{5}$. If the condition were $\le \sqrt{5}$, we would include the endpoints where the difference is exactly $\sqrt{5}$. Since it is a strict inequality, we must use an open interval to exclude the cases where the difference equals $\sqrt{5}$.

Q: Could I have used the discriminant formula directly to solve this?

Absolutely! The difference of the roots of a quadratic $Ax^2 + Bx + C = 0$ is given by $\frac{\sqrt{D}}{|A|}$, where $D = B^2 - 4AC$. In your equation, $A=1$, $B=a$, and $C=1$, so the difference is $\sqrt{a^2 - 4}$. Setting $\sqrt{a^2 - 4} < \sqrt{5}$ and squaring both sides gives $a^2 - 4 < 5$, which leads to the exact same result. Both methods are perfectly valid!

Select Answer:

Visualized Solution

The Quadratic Equation

Given equation: $x^{2} + a x + 1 = 0$
Let the roots of this equation be $α$ and $β$ .

The Difference of Roots

The distance between the roots is $∣ α - β ∣$ .
Given condition: $∣ α - β ∣ < 5$ .

Squaring the Condition

To remove the modulus and square root, we square both sides.
$(α - β)^{2} < 5$

Algebraic Identity

We know the identity: $(α - β)^{2} = (α + β)^{2} - 4 α β$
This connects the difference of roots to their sum and product.

Sum and Product of Roots

From $x^{2} + a x + 1 = 0$ :
Sum of roots: $α + β = - a$
Product of roots: $α β = 1$

Substituting into the Inequality

Substitute $α + β = - a$ and $α β = 1$ into the identity.
$(- a)^{2} - 4 (1) < 5$

Simplifying the Expression

Simplify the squared term and the product.
$a^{2} - 4 < 5$

Solving for $a^{2}$

Add $4$ to both sides of the inequality.
$a^{2} < 5 + 4$
$a^{2} < 9$

Finding the Range of $a$

The inequality $a^{2} < 9$ implies $∣ a ∣ < 3$ .
This means $a$ must lie strictly between $- 3$ and $3$ .
Final Answer: $a \in (- 3, 3)$

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The Sigma Insight: Relation Between Roots and Coefficients

Solution Diagram

Analyzing the Setup

We are investigating the quadratic equation

x^{2} + a x + 1 = 0

. The roots of this equation,

α

and

β

, represent the points where the parabola intersects the x-axis.

The problem imposes a specific geometric constraint: the distance between these roots must be less than

5

. Mathematically, this is expressed as

∣ α - β ∣ < 5

The Geometric Vision

To simplify the expression

∣ α - β ∣ < 5

, we can square both sides of the inequality. Since both sides are non-negative, the inequality remains valid:

(α - β)^{2} < 5

This transformation removes the absolute value and the square root, providing a much more manageable algebraic form.

The Algebraic Bridge

We utilize Vieta's formulas to relate the roots to the coefficients of the quadratic equation

x^{2} + a x + 1 = 0

. For this equation, the sum and product of the roots are:

α + β = - a

α β = 1

We connect these values to the squared difference using the fundamental algebraic identity:

(α - β)^{2} = (α + β)^{2} - 4 α β

The Final Inequality

Substituting our known values into the identity, we replace

(α + β)

with

- a

and

(α β)

with

1

(- a)^{2} - 4 (1) < 5

Simplifying this expression leads to:

a^{2} - 4 < 5

a^{2} < 9

This inequality implies that the absolute value of

a

must be less than

3

, or

∣ a ∣ < 3

. Therefore, the set of possible values for

a

is the interval:

$a \in (- 3, 3)$

Common Traps & FAQs

Why do we square the inequality

∣ α - β ∣ < 5

? Can't we just work with the modulus directly?

Working with the modulus is certainly possible, but it often leads to complex case-based analysis. Squaring both sides is a powerful, elegant technique here because both sides are positive, which ensures the inequality sign remains unchanged. It effectively removes the square root and the modulus, simplifying the expression into a straightforward quadratic inequality.

Why is the final interval

(- 3, 3)

open? Shouldn't it be closed?

That is a great observation! The problem specifies that the difference between the roots is *strictly* less than

5

. If the condition were

\leq 5

, we would include the endpoints where the difference is exactly

5

. Since it is a strict inequality, we must use an open interval to exclude the cases where the difference equals

5

Could I have used the discriminant formula directly to solve this?

Absolutely! The difference of the roots of a quadratic

A x^{2} + B x + C = 0

is given by

\frac{D}{∣ A ∣}

, where

D = B^{2} - 4 A C

. In your equation,

A = 1

B = a

, and

C = 1

, so the difference is

a^{2} - 4

. Setting

a^{2} - 4 < 5

and squaring both sides gives

a^{2} - 4 < 5

, which leads to the exact same result. Both methods are perfectly valid!

Animated Solution for Mathematics - Quadratic Equations: If the difference between the roots of the equation $x^{2} + a x + 1 = 0$ is less than $5$ , then the set of possible values of $a$ is

Select Answer:

Visualized Solution

The Quadratic Equation

The Difference of Roots

Squaring the Condition

Algebraic Identity

Sum and Product of Roots

Substituting into the Inequality

Simplifying the Expression

Solving for $a^{2}$

Finding the Range of $a$

The Sigma Insight: Relation Between Roots and Coefficients

Analyzing the Setup

The Geometric Vision

The Algebraic Bridge

The Final Inequality

Similar Questions

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Select Answer:

Visualized Solution

The Quadratic Equation

The Difference of Roots

Squaring the Condition

Algebraic Identity

Sum and Product of Roots

Substituting into the Inequality

Simplifying the Expression

Solving for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } a2

Finding the Range of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } a

The Sigma Insight: Relation Between Roots and Coefficients

Analyzing the Setup

The Geometric Vision

The Algebraic Bridge

The Final Inequality

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Solving for $a^{2}$

Finding the Range of $a$

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