(A) For a quadratic equation of the form $ax^{2} + bx + c = 0$,the roots are equal and real if the discriminant $D = b^{2} - 4ac = 0$.Given equation:

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(A) For a quadratic equation of the form $ax^{2} + bx + c = 0$,the roots are equal and real if the discriminant $D = b^{2} - 4ac = 0$.Given equation: $k x^{2} - 2 \sqrt{5} x + 4 = 0$.Here,$a = k$,$b = -2 \sqrt{5}$,and $c = 4$.Substituting these values into the discriminant formula:$(-2 \sqrt{5})^{2} - 4(k)(4) = 0$$(4 \times 5) - 16k = 0$$20 - 16k = 0$$16k = 20$$k = \frac{20}{16} = \frac{5}{4}$.

Class 10 Mathematics Quadratic Equations Nature of Roots

Medium

If the following quadratic equation has two equal and real roots,then find the value of $k$: $k x^{2} - 2 \sqrt{5} x + 4 = 0$.

A
$\frac{5}{4}$
B
$\frac{15}{3}$
C
$\frac{4}{1}$
D
$\frac{10}{3}$

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Find the discriminant of the following quadratic equation and hence determine the nature of the roots of the equation: $4x^{2} + 11x + 10 = 0$

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If the following quadratic equation has two equal and real roots,then find the value of $k$: $3x^{2} - 18x + k = 0$.

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Which of the following equations has the sum of its roots as $3$?

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Find the nature of the roots of the following quadratic equation. If real roots exist,find them:
$3x^2 - 4\sqrt{3}x + 4 = 0$

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State whether the following statements are true or false. Justify your answers.
$(i)$ Every quadratic equation has exactly one root.
$(ii)$ Every quadratic equation has at least one real root.

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If the following quadratic equation has two equal and real roots,then find the value of $k$: $k x^{2} - 2 \sqrt{5} x + 4 = 0$.

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Find the discriminant of the following quadratic equation and hence determine the nature of the roots of the equation: $4x^{2} + 11x + 10 = 0$

If the following quadratic equation has two equal and real roots,then find the value of $k$: $3x^{2} - 18x + k = 0$.

Which of the following equations has the sum of its roots as $3$?

Find the nature of the roots of the following quadratic equation. If real roots exist,find them:$3x^2 - 4\sqrt{3}x + 4 = 0$

State whether the following statements are true or false. Justify your answers.$(i)$ Every quadratic equation has exactly one root.$(ii)$ Every quadratic equation has at least one real root.

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Find the nature of the roots of the following quadratic equation. If real roots exist,find them:
$3x^2 - 4\sqrt{3}x + 4 = 0$

State whether the following statements are true or false. Justify your answers.
$(i)$ Every quadratic equation has exactly one root.
$(ii)$ Every quadratic equation has at least one real root.