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Quadratic Equations — Maths STD 10 — Question

CBSE BoardEnglish MediumSTD 10MathsQuadratic Equations2 Marks
Question
State whether the following quadratic equations have two distinct real roots. Justify your answer.
$\sqrt{2}\text{x}^2-\frac{3}{\sqrt{2}}\text{x}+\frac{1}{\sqrt{2}}=0$
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Answer

Main concept used:
Quadratic equation $ax^2 + bx + c = 0$ will have two distinct real roots if $D > 0 or b^2 - 4ac > 0.$
$\sqrt{2}\text{x}^2-\frac{3}{\sqrt{2}}\text{x}+\frac{1}{\sqrt{2}}=0$
$\text{D}=\text{b}^2-4\text{ac}$
$\Rightarrow\ \text{D}=\Big(\frac{-3}{\sqrt{2}}\Big)^2-4(\sqrt{2})\Big(\frac{1}{\sqrt{2}}\Big)$
$\Big(\text{a}=\sqrt{2},\text{ b}=\frac{-3}{\sqrt{2}},\text{ c}=+\frac{1}{\sqrt{2}}\Big)$
$=\frac{9}{2}-\frac{4}{1}=\frac{9-8}{2}$
$\Rightarrow\ \text{D}=\frac{1}{2}>0$
$\therefore\ \text{D}>0$
Hence, the given quadratic equation has two real and distirict roots.

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