Question

one of the root of equation
$kx {}^{2} - 3x - 1 = 0 \: is \: \frac{1}{2}$ solve the following to find the value of k (x=½)

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Answer 1

Step by step explanation:-

Given :-

For the equation kx² - 3x - 1 = 0 And Also given 1/2 is a factor that means x = 1/2

To find :-

Value of k

Solution:-

Hence It is a root If we Substitute x value It should be equal to 0 So, Substitute x = 1/2 to get value of k

kx² - 3x - 1 = 0

$k( \dfrac{1}{2} ) {}^{2} - 3( \dfrac{1}{2} ) - 1 = 0$

$k ( \dfrac{1}{4} ) - \dfrac{3}{2} - 1 = 0$

$\dfrac{k}{4} - \dfrac{3}{2} - 1 = 0$

Taking LCM To denominator

LCM of 4,2 is 2

$\dfrac{k - 2(3) - 1(4)}{4} = 0$

$\dfrac{k - 6 - 4}{4} = 0$

$k - 10 = 0(4)$

k - 10 = 0

k = 10

Verification:-

Substitute value of k, x it should be equal to 0

kx² - 3x - 1 = 0

10 (1/2)² - 3 $\sf\dfrac{1}{2}$ -1 = 0

10 $\sf\dfrac{1}{4}$ - $\sf\dfrac{3}{2}$- 1 = 0

$\sf\dfrac{10}{4}$ - $\sf\dfrac{3}{2}$- 1 = 0

$\sf\dfrac{5}{2}$ - $\sf\dfrac{3}{2}$ -1 = 0

$\sf\dfrac{5-3 -2}{2}$ = 0

$\sf\dfrac{5-5} {2}$=0

$\sf\dfrac{0}{2}$ =0

0=0

LHS = RHS

Verified

Answer 2

Given : A quadratic equation kx² - 3x - 1 = 0 and one of the zero is 1/2 .

To Find : Value of k.

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To find the value of k, firstly substitute the value of (x = 1/2) in the equation.

$\: \:$

$\mathfrak{\underline{ \bigstar \: substituting \: the \: values \: : }}$

$\sf : \implies \: kx {}^{2} - 3x - 1 = 0 \\ \\ \\ \sf : \implies \: k\left( \frac{1}{2 } \right) {}^{2} - 3 \left( \frac{1}{2} \right) - 1 = 0 \\ \\ \\ \sf : \implies \: \frac{k}{4} - \frac{3}{2} - 1 = 0 \\ \\ \\ \sf : \implies \: \frac{k - 6 - 4}{4} = 0 \\ \\ \\ \sf : \implies \: \frac{k - 10}{4} = 0 \\ \\ \\ \sf : \implies \: k - 10 = 0 \\ \\ \\ \sf : \implies\underline{\boxed{\mathfrak\purple{k = 10}}} \: \bigstar \\ \\ \\ \\ \sf\therefore{\underline{Hence, \: the \: value \: of \: k \: is \: \bold{10}.}}$

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$\: \:$

$\mathfrak{ \underline{ \bigstar \: Verification \: : }}$

$\:$

$\sf:\implies \: kx {}^{2} - 3x - 1 = 0 \\ \\ \\ \sf:\implies \:10\left( \frac{1}{2} \right) {}^{2} - 3\left( \frac{1}{2} \right) - 1 = 0 \\ \\ \\ \sf:\implies \frac{10}{4} - \frac{3}{2} - 1 = 0 \\ \\ \\ \sf:\implies \frac{10 - 6 - 4}{10} = 0 \\ \\ \\ \sf:\implies \frac{10 - 10}{10} = 0 \\ \\ \\ \sf:\implies0 = 0 \\ \\ \\ \\ \sf\therefore{\underline{Hence, verified.}}$