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Animated Solution for Mathematics - Quadratic Equations: Let two numbers have arithmetic mean 9 and geometric mean 4. Then these numbers are the roots of the quadratic equation

Q: What if the GM was negative? Does the formula still hold?

The Geometric Mean of two positive numbers is defined as $\sqrt{ab}$, which is always non-negative. If the problem gave a negative GM, it would imply the numbers themselves might be complex or negative, which would change the nature of the roots. Always check the domain!

Select Answer:

Visualized Solution

Defining the Roots $a$ and $b$

Let the two unknown numbers be $a$ and $b$ .
These numbers will act as the roots of our required quadratic equation.

Arithmetic Mean ( $A M$ )

The Arithmetic Mean ( $A M$ ) of two numbers $a$ and $b$ is their average.
Formula: $A M = \frac{a + b}{2}$

Substituting the $A M$ Value

We are given that $A M = 9$ .
Substituting this into our formula:
$\frac{a + b}{2} = 9$

Calculating the Sum of Roots ( $S$ )

Multiply both sides by $2$ to isolate $a + b$ .
$a + b = 9 \times 2$
Sum of roots ( $S$ ) = $18$

Geometric Mean ( $GM$ )

The Geometric Mean ( $GM$ ) of two positive numbers $a$ and $b$ is the square root of their product.
Formula: $GM = ab$

Substituting the $GM$ Value

We are given that $GM = 4$ .
Substituting this into our formula:
$ab = 4$

Calculating the Product of Roots ( $P$ )

To remove the square root, square both sides of the equation.
$(ab)^{2} = 4^{2}$
Product of roots ( $P$ ) = $16$

Quadratic Equation Template

A quadratic equation can be formed if the sum ( $S$ ) and product ( $P$ ) of its roots are known.
Standard Template: $x^{2} - (Sum) x + (Product) = 0$
$x^{2} - S x + P = 0$

Final Substitution

We found $S = 18$ and $P = 16$ .
Substitute these values into the template:
$x^{2} - 18 x + 16 = 0$

Conclusion

The required quadratic equation is $x^{2} - 18 x + 16 = 0$ .
Comparing with the given options, this matches Option 2.

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The Sigma Insight: Relation Between Roots and Coefficients

Solution Diagram

Analyzing the Setup

To solve for the quadratic equation, we first identify the properties of the roots

a

and

b

using the given Arithmetic Mean (AM) and Geometric Mean (GM).

The Arithmetic Mean is defined as:

\frac{a + b}{2} = 9

This immediately reveals that the sum of the roots,

S = a + b

, is equal to

18

The Master Equation

Next, we utilize the Geometric Mean, which is defined as:

ab = 4

By squaring both sides, we determine the product of the roots,

P = ab

, to be

16

We now invoke the power of Vieta's formulas. Any quadratic equation with roots

a

and

b

follows the standard template:

x^{2} - (a + b) x + ab = 0

Final Calculation

By substituting our calculated values for the sum and product into the template, we obtain the final equation:

x^{2} - 18 x + 16 = 0

The beauty of this approach lies in the fact that we never needed to calculate the individual values of

a

b

. We only required their collective properties to define the structure of the equation.

Strategic Takeaway

In the JEE examination, time is your most precious resource. By mastering the relationship between roots and coefficients, you bypass tedious calculations and move directly to the solution.

Always look for the underlying structure, trust the formulas, and aim for the most elegant path. The equation $x^{2} - 18 x + 16 = 0$ is the perfect encapsulation of the relationship between these two mysterious roots.

Common Traps & FAQs

Wait, why is the formula

x^{2} - S x + P = 0

and not

x^{2} + S x + P = 0

That is a fantastic question! Remember Vieta's formulas. For a quadratic equation

a x^{2} + b x + c = 0

, the sum of the roots is

- b / a

. If we normalize the equation to

x^{2} + (b / a) x + (c / a) = 0

, the coefficient of

x

is the negative of the sum of the roots. Hence, the negative sign is essential!

Do I need to solve for

a

and

b

individually?

Not at all! While you could solve the system of equations, it is an unnecessary detour. The beauty of quadratic theory is that we can construct the equation directly from the sum and product of the roots. Efficiency is key in JEE!

What if the GM was negative? Does the formula still hold?

The Geometric Mean of two positive numbers is defined as

ab

, which is always non-negative. If the problem gave a negative GM, it would imply the numbers themselves might be complex or negative, which would change the nature of the roots. Always check the domain!

Animated Solution for Mathematics - Quadratic Equations: Let two numbers have arithmetic mean 9 and geometric mean 4. Then these numbers are the roots of the quadratic equation

Select Answer:

Visualized Solution

Defining the Roots $a$ and $b$

Arithmetic Mean ( $A M$ )

Substituting the $A M$ Value

Calculating the Sum of Roots ( $S$ )

Geometric Mean ( $GM$ )

Substituting the $GM$ Value

Calculating the Product of Roots ( $P$ )

Quadratic Equation Template

Final Substitution

Conclusion

The Sigma Insight: Relation Between Roots and Coefficients

Analyzing the Setup

The Master Equation

Final Calculation

Strategic Takeaway

Similar Questions

Let $p$ and $q$ be roots of the equation $x^{2} - A x + A = 0$ and let $r$ and $s$ be the roots of the equation $x^{2} - 18 x + B = 0$ . If $p < q < r < s$ are in arithmetic progression, then $A = \dots$ and $B = \dots$ .

Difference between the corresponding roots of $x^{2} + a x + b = 0$ and $x^{2} + b x + a = 0$ is same and $a \neq = b$ , then

The harmonic mean of the roots of the equation $(5 + 2) x^{2} - (4 + 5) x + 8 + 25 = 0$ is

If the sum of the roots of the quadratic equation $a x^{2} + b x + c = 0$ is equal to the sum of the squares of their reciprocals, then $a / c, b / a$ and $c / b$ are in

Let $α, β$ be the roots of the equation $x^{2} + 22 x - 1 = 0$ . The quadratic equation, whose roots are $α^{4} + β^{4}$ and $\frac{1}{10} (α^{6} + β^{6})$ , is :

Let $p$ and $q$ be real numbers such that $p \neq = 0, p^{3} \neq = - q$ . If $α$ and $β$ are nonzero complex numbers satisfying $α + β = - p$ and $α^{3} + β^{3} = q$ , then a quadratic equation having $α / β$ and $β / α$ as its roots is

Let p, q and r be real numbers ( $p \neq = q, r \neq = 0$ ), such that the roots of the equation $\frac{1}{x + p} + \frac{1}{x + q} = \frac{1}{r}$ are equal in magnitude but opposite in sign, then the sum of squares of these roots is equal to :

Let $α$ and $β$ be the roots of equation $p x^{2} + q x + r = 0, p \neq = 0$ . If $p, q, r$ are in A.P. and $1/ α + 1/ β = 4$ , then the value of $∣ α - β ∣$ is:

Let $a, b \in R$ be such that the equation $a x^{2} - 2 b x + 15 = 0$ has a repeated root $α$ . If $α$ and $β$ are the roots of the equation $x^{2} - 2 b x + 21 = 0$ , then $α^{2} + β^{2}$ is equal to:

If $α, β$ , where $α < β$ , are the roots of the equation $λ x^{2} - (λ + 3) x + 3 = 0$ such that $\frac{1}{α} - \frac{1}{β} = \frac{1}{3}$ , then the sum of all possible values of $λ$ is

Animated Solution for Mathematics - Quadratic Equations: Let two numbers have arithmetic mean 9 and geometric mean 4. Then these numbers are the roots of the quadratic equation

Select Answer:

Visualized Solution

Defining the Roots .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } a and .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } b

Arithmetic Mean ( .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } AM)

Substituting the .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } AM Value

Calculating the Sum of Roots ( .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } S)

Geometric Mean ( .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } GM)

Substituting the .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } GM Value

Calculating the Product of Roots ( .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } P)

Quadratic Equation Template

Final Substitution

Conclusion

The Sigma Insight: Relation Between Roots and Coefficients

Analyzing the Setup

The Master Equation

Final Calculation

Strategic Takeaway

Similar Questions

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let p and q be roots of the equation x2−Ax+A=0 and let r and s be the roots of the equation x2−18x+B=0. If p<q<r<s are in arithmetic progression, then A=… and B=….

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Difference between the corresponding roots of x2+ax+b=0 and x2+bx+a=0 is same and a=b, then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The harmonic mean of the roots of the equation (5+2​)x2−(4+5​)x+8+25​=0 is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the sum of the roots of the quadratic equation ax2+bx+c=0 is equal to the sum of the squares of their reciprocals, then a/c,b/a and c/b are in

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let α,β be the roots of the equation x2+22​x−1=0. The quadratic equation, whose roots are α4+β4 and 101​(α6+β6), is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let α and β be the roots of equation px2+qx+r=0,p=0. If p,q,r are in A.P. and 1/α+1/β=4, then the value of ∣α−β∣ is:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let a,b∈R be such that the equation ax2−2bx+15=0 has a repeated root α. If α and β are the roots of the equation x2−2bx+21=0, then α2+β2 is equal to:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If α,β, where α<β, are the roots of the equation λx2−(λ+3)x+3=0 such that α1​−β1​=31​, then the sum of all possible values of λ is

Defining the Roots $a$ and $b$

Arithmetic Mean ( $A M$ )

Substituting the $A M$ Value

Calculating the Sum of Roots ( $S$ )

Geometric Mean ( $GM$ )

Substituting the $GM$ Value

Calculating the Product of Roots ( $P$ )

Let $p$ and $q$ be roots of the equation $x^{2} - A x + A = 0$ and let $r$ and $s$ be the roots of the equation $x^{2} - 18 x + B = 0$ . If $p < q < r < s$ are in arithmetic progression, then $A = \dots$ and $B = \dots$ .

Difference between the corresponding roots of $x^{2} + a x + b = 0$ and $x^{2} + b x + a = 0$ is same and $a \neq = b$ , then

The harmonic mean of the roots of the equation $(5 + 2) x^{2} - (4 + 5) x + 8 + 25 = 0$ is

If the sum of the roots of the quadratic equation $a x^{2} + b x + c = 0$ is equal to the sum of the squares of their reciprocals, then $a / c, b / a$ and $c / b$ are in

Let $α, β$ be the roots of the equation $x^{2} + 22 x - 1 = 0$ . The quadratic equation, whose roots are $α^{4} + β^{4}$ and $\frac{1}{10} (α^{6} + β^{6})$ , is :

Let $α$ and $β$ be the roots of equation $p x^{2} + q x + r = 0, p \neq = 0$ . If $p, q, r$ are in A.P. and $1/ α + 1/ β = 4$ , then the value of $∣ α - β ∣$ is:

Let $a, b \in R$ be such that the equation $a x^{2} - 2 b x + 15 = 0$ has a repeated root $α$ . If $α$ and $β$ are the roots of the equation $x^{2} - 2 b x + 21 = 0$ , then $α^{2} + β^{2}$ is equal to:

If $α, β$ , where $α < β$ , are the roots of the equation $λ x^{2} - (λ + 3) x + 3 = 0$ such that $\frac{1}{α} - \frac{1}{β} = \frac{1}{3}$ , then the sum of all possible values of $λ$ is