First slide

Theory of equations

Question

If x₁ and x₂ are the arithmetic and harmonic means of the roots of the equation ${ax}^{2} + bx + c = 0$ , the equation whose quadratic roots are x₁ and x₂, is

Moderate

A

${abx}^{2} + (b^{2} + ac) x + bc = 0$
B

$2 {abx}^{2} + (b^{2} + 4 ac) x + 2 bc = 0$
C

$2 {abx}^{2} + (b^{2} + ac) x + bc = 0$
D

None of the above

Solution

Let $α$ and $β$ be the roots of ${ax}^{2} + bx + c = 0$
$\begin{aligned} x_{1} = \frac{α + β}{2} = - \frac{b}{2 a} \\ x_{2} = \frac{2 αβ}{α + β} = \frac{2 \cdot \frac{c}{a}}{- \frac{b}{a}} = - \frac{2 c}{b} \end{aligned}$
The required equation is
$\begin{array}{r} x^{2} - [(- \frac{b}{2 a}) + (- \frac{2 c}{b})] x + \frac{2 bc}{2 ab} = 0 \\ 2 {abx}^{2} + (b^{2} + 4 ac) x + 2 bc = 0 \end{array}$

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