(A) For a quadratic equation $ax^2 + bx + c = 0$ to have two equal and real roots,the discriminant $D = b^2 - 4ac$ must be equal to $0$.Here,$a = (k+1

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(A) For a quadratic equation $ax^2 + bx + c = 0$ to have two equal and real roots,the discriminant $D = b^2 - 4ac$ must be equal to $0$.Here,$a = (k+1)$,$b = -2(k-1)$,and $c = 1$.Substituting these values into the discriminant formula:$D = [-2(k-1)]^2 - 4(k+1)(1) = 0$$4(k-1)^2 - 4(k+1) = 0$Dividing by $4$:$(k-1)^2 - (k+1) = 0$$k^2 - 2k + 1 - k - 1 = 0$$k^2 - 3k = 0$$k(k-3) = 0$Therefore,$k = 0$ or $k = 3$.

Class 10 Mathematics Quadratic Equations Mix Examples - Quadratic Equations

Medium

If the following quadratic equation has two equal and real roots,then find the value of $k$: $(k+1)x^2 - 2(k-1)x + 1 = 0$.

A
$0$ and $3$
B
$1$ and $2$
C
$7$ and $-3$
D
$3$ and $-4$

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Mix Examples - Quadratic Equations

Solve the following equation using the method of factorization and write its solution set: $x - \frac{1}{x} - \frac{45}{14} = 0$; $x \neq 0$.

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The product of the digits of a two-digit number is $14$. The number obtained by interchanging the digits is $45$ more than the original number. Find the original number.

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If the following quadratic equation has two equal and real roots,then find the value of $k$: $16x^{2} - 40x + \frac{k-1}{2} = 0$.

Medium

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Find the roots of the quadratic equation by using the quadratic formula: $-x^{2}+7x-10=0$.

Medium

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Find the discriminant of the following quadratic equation and hence determine the nature of the roots of the equation: $3x^{2} - 18x + 27 = 0$.

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If the following quadratic equation has two equal and real roots,then find the value of $k$: $(k+1)x^2 - 2(k-1)x + 1 = 0$.

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Solve the following equation using the method of factorization and write its solution set: $x - \frac{1}{x} - \frac{45}{14} = 0$; $x \neq 0$.

The product of the digits of a two-digit number is $14$. The number obtained by interchanging the digits is $45$ more than the original number. Find the original number.

If the following quadratic equation has two equal and real roots,then find the value of $k$: $16x^{2} - 40x + \frac{k-1}{2} = 0$.

Find the roots of the quadratic equation by using the quadratic formula: $-x^{2}+7x-10=0$.

Find the discriminant of the following quadratic equation and hence determine the nature of the roots of the equation: $3x^{2} - 18x + 27 = 0$.

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