In each of the following, determine whether the given numbers are roots of the given equations or not; (i) \[{{\mathbf{x}}^{\mathbf{2}}}~\text{ }\mathbf{5x}\text{ }+\text{ }\mathbf{6}\text{ }=\text{ }\mathbf{0};\text{ }\mathbf{2},\text{ }-\mathbf{3}\] (ii) \[\mathbf{3}{{\mathbf{x}}^{\mathbf{2}}}~\text{ }\mathbf{13x}\text{ }\text{ }\mathbf{10}\text{ }=\text{ }\mathbf{0};\text{ }\mathbf{5},\text{ }-\mathbf{2}/\mathbf{3}\]

In each of the following, determine whether the given numbers are roots of the given equations or not; (i)

${{\mathbf{x}}^{\mathbf{2}}}~\text{ }\mathbf{5x}\text{ }+\text{ }\mathbf{6}\text{ }=\text{ }\mathbf{0};\text{ }\mathbf{2},\text{ }-\mathbf{3}$

(ii)

$\mathbf{3}{{\mathbf{x}}^{\mathbf{2}}}~\text{ }\mathbf{13x}\text{ }\text{ }\mathbf{10}\text{ }=\text{ }\mathbf{0};\text{ }\mathbf{5},\text{ }-\mathbf{2}/\mathbf{3}$

In each of the following, determine whether the given numbers are roots of the given equations or not; (i)

${{\mathbf{x}}^{\mathbf{2}}}~\text{ }\mathbf{5x}\text{ }+\text{ }\mathbf{6}\text{ }=\text{ }\mathbf{0};\text{ }\mathbf{2},\text{ }-\mathbf{3}$

(ii)

$\mathbf{3}{{\mathbf{x}}^{\mathbf{2}}}~\text{ }\mathbf{13x}\text{ }\text{ }\mathbf{10}\text{ }=\text{ }\mathbf{0};\text{ }\mathbf{5},\text{ }-\mathbf{2}/\mathbf{3}$

(i)

${{\mathbf{x}}^{\mathbf{2}}}~\text{ }\mathbf{5x}\text{ }+\text{ }\mathbf{6}\text{ }=\text{ }\mathbf{0};\text{ }\mathbf{2},\text{ }-\mathbf{3}$

Let us substitute the given values in the expression and check,

When,

$\begin{array}{*{35}{l}} x\text{ }=\text{ }2 \\ {{x}^{2}}~\text{ }5x\text{ }+\text{ }6\text{ }=\text{ }0 \\ {{\left( 2 \right)}^{2}}~\text{ }5\left( 2 \right)\text{ }+\text{ }6\text{ }=\text{ }0 \\ 4\text{ }\text{ }10\text{ }+\text{ }6\text{ }=\text{ }0 \\ 0\text{ }=\text{ }0 \\ \therefore x\text{ }=\text{ }0 \\ \end{array}$

When,

$\begin{array}{*{35}{l}} x\text{ }=\text{ }-3 \\ {{x}^{2}}~\text{ }5x\text{ }+\text{ }6\text{ }=\text{ }0 \\ {{\left( -3 \right)}^{2}}~\text{ }5\left( -3 \right)\text{ }+\text{ }6\text{ }=\text{ }0 \\ 9\text{ }+\text{ }15\text{ }+\text{ }6\text{ }=\text{ }0 \\ 30\text{ }=\text{ }0 \\ \therefore x\text{ }\ne \text{ }0 \\ \end{array}$

Hence, the value

$x\text{ }=2$

is the root of the equation.

And value

$x\text{ }=\text{ }-3$

is not a root of the equation.

(ii)

$\mathbf{3}{{\mathbf{x}}^{\mathbf{2}}}~\text{ }\mathbf{13x}\text{ }\text{ }\mathbf{10}\text{ }=\text{ }\mathbf{0};\text{ }\mathbf{5},\text{ }-\mathbf{2}/\mathbf{3}$

Let us substitute the given values in the expression and check,

When,

$\begin{array}{*{35}{l}} x\text{ }=\text{ }5 \\ 3{{x}^{2}}~\text{ }13x\text{ }\text{ }10\text{ }=\text{ }0 \\ 3{{\left( 5 \right)}^{2}}~\text{ }13\left( 5 \right)\text{ }\text{ }10\text{ }=\text{ }0 \\ 3\left( 25 \right)\text{ }\text{ }65\text{ }\text{ }10\text{ }=\text{ }0 \\ 75\text{ }\text{ }75\text{ }=\text{ }0 \\ 0\text{ }=\text{ }0 \\ \therefore x\text{ }=\text{ }0 \\ \end{array}$

When,

$\begin{array}{*{35}{l}} x\text{ }=\text{ }-2/3 \\ 3{{x}^{2}}~\text{ }13x\text{ }\text{ }10\text{ }=\text{ }0 \\ 3{{\left( -2/3 \right)}^{2}}~\text{ }13\left( -2/3 \right)\text{ }\text{ }10\text{ }=\text{ }0 \\ 4/9\text{ }+\text{ }26/3\text{ }\text{ }10\text{ }=\text{ }0 \\ 4/3\text{ }+\text{ }26/3\text{ }\text{ }10\text{ }=\text{ }0 \\ 30/3\text{ }\text{ }10\text{ }=\text{ }0 \\ 10\text{ }\text{ }10\text{ }=\text{ }0 \\ \therefore x\text{ }=\text{ }0 \\ \end{array}$

Hence, the value

$x\text{ }=\text{ }5,\text{ }-2/3$

are the roots of the equation.

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