(D) 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 the equati

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(D) 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 the equation $4x^{2} - 5x + k = 0$,we have $a = 4$,$b = -5$,and $c = k$.Substituting these values into the discriminant formula:$D = (-5)^{2} - 4(4)(k) = 0$$25 - 16k = 0$$16k = 25$$k = \frac{25}{16}$

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 $k$: $4x^{2} - 5x + k = 0$.

A
$\frac{25}{49}$
B
$\frac{5}{15}$
C
$\frac{13}{10}$
D
$\frac{25}{16}$

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Quadratic Equations

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Nature of Roots

State whether the quadratic equation $\sqrt{2} x^{2}-\frac{3}{\sqrt{2}} x + \frac{1}{\sqrt{2}} = 0$ has two distinct real roots. Justify your answer.

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$(x^{2}+1)^{2}-x^{2}=0$ has

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If the following quadratic equation has two equal and real roots,then find the value of $k$: $5x^{2} - 2kx + 20 = 0$.

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Find the roots of the quadratic equation by applying the quadratic formula: $4 x^{2}+4 \sqrt{3} x+3=0$.

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If both the roots of a quadratic equation $ax^2 + bx + c = 0$ $(a \neq 0)$ are equal,then ..... .

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

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State whether the quadratic equation $\sqrt{2} x^{2}-\frac{3}{\sqrt{2}} x + \frac{1}{\sqrt{2}} = 0$ has two distinct real roots. Justify your answer.

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If the following quadratic equation has two equal and real roots,then find the value of $k$: $5x^{2} - 2kx + 20 = 0$.

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