If 2+i√3 is a root of the equation x² +...

MathematicsRelation Between Roots and CoefficientsJEE Advanced 1982Easy

If $2 + i 3$ is a root of the equation $x^{2} + p x + q = 0$ , where $p$ and $q$ are real, then $(p, q) = (\dots, \dots)$ .

Answer:(-4, 7)

Visualized Solution (English)

Visualizing the Root .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } 2 + i 3

Given root: $α = 2 + i 3$
Equation: $x^{2} + p x + q = 0$
Coefficients $p, q \in R$

Complex Conjugate Root Theorem

For a quadratic equation $a x^{2} + b x + c = 0$
If $a, b, c \in R$ , then complex roots occur in conjugate pairs.
If $α = a + ib$ is a root, then $\overset{α}{ˉ} = a - ib$ is also a root.

Identifying the Second Root

First root: $α = 2 + i 3$
Second root: $β = \overset{α}{ˉ} = 2 - i 3$

Vieta's Formulas

For $x^{2} + p x + q = 0$ :
Sum of roots: $α + β = - p$
Product of roots: $α β = q$

Calculating the Sum of Roots

Sum $= (2 + i 3) + (2 - i 3)$
Sum $= 2 + 2 + i 3 - i 3$
Sum $= 4$

Calculating the Product of Roots

Product $= (2 + i 3) (2 - i 3)$
Product $= 2^{2} - (i 3)^{2}$
Product $= 4 - (- 3)$
Product $= 7$

Finding .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } p and .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } q

$- p = 4 ⟹ p = - 4$
Product $= q = 7$
$(p, q) = (- 4, 7)$

Final Result and Takeaway

Final Answer: $(p, q) = (- 4, 7)$
Key Takeaway: Complex roots of equations with real coefficients always come in conjugate pairs.
Next Challenge: What if the coefficients $p$ and $q$ were not real?

00:00 / 00:00

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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

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If $p, q, r$ are +ve and are in A.P., the roots of quadratic equation $p x^{2} + q x + r = 0$ are all real for

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Given that $x = - 9$ is a root of $x 27 3 x 6 72 x = 0$ the other two roots are ......... and .........

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Let $a, b, c$ be real numbers with $a \neq = 0$ and let $α, β$ be the roots of the equation $a x^{2} + b x + c = 0$ . Express the roots of $a^{3} x^{2} + ab c x + c^{3} = 0$ in terms of $α, β$ .

Answer:

α^{2} β, α β^{2}

If $2 + i 3$ is a root of the equation $x^{2} + p x + q = 0$ , where $p$ and $q$ are real, then $(p, q) = (\dots, \dots)$ .

Visualized Solution (English)

Visualizing the Root $2 + i 3$

Complex Conjugate Root Theorem

Identifying the Second Root

Vieta's Formulas

Calculating the Sum of Roots

Calculating the Product of Roots

Finding $p$ and $q$

Final Result and Takeaway

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.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If 2+i3​ is a root of the equation x2+px+q=0, where p and q are real, then (p,q)=(…,…).

Visualized Solution (English)

Visualizing the Root .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } 2+i3​

Complex Conjugate Root Theorem

Identifying the Second Root

Vieta's Formulas

Calculating the Sum of Roots

Calculating the Product of Roots

Finding .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } p and .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } q

Final Result and Takeaway

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If $2 + i 3$ is a root of the equation $x^{2} + p x + q = 0$ , where $p$ and $q$ are real, then $(p, q) = (\dots, \dots)$ .

Visualizing the Root $2 + i 3$

Finding $p$ and $q$

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