Solve the following equation and give your answer correct to 2 decimal places 5x 2 4 0

(i) \[\mathbf{x}{}^\text{2}\text{ }\text{ }\mathbf{5x}\text{ }\text{ }\mathbf{10}\text{ }=\text{ }\mathbf{0}\]

Let us consider,

\[a\text{ }=\text{ }1,\text{ }b\text{ }=\text{ }-5,\text{ }c\text{ }=\text{ }-10\]

So, by using the formula,

\[\begin{array}{*{35}{l}}

D\text{ }=\text{ }{{b}^{2}}~\text{ }4ac  \\

=\text{ }{{\left( -5 \right)}^{2}}~\text{ }4\left( 1 \right)\text{ }\left( -10 \right)  \\

=\text{ }25\text{ }+\text{ }40  \\

=\text{ }65  \\

\end{array}\]

So,

\[\begin{array}{*{35}{l}}

x\text{ }=\text{ }\left[ -\left( -5 \right)~\pm \text{ }\surd 65 \right]\text{ }/\text{ }2\left( 1 \right)  \\

=\text{ }\left[ 5~\pm \text{ }\surd 65 \right]\text{ }/\text{ }2  \\

=\text{ }\left[ 5~\pm ~8.06 \right]\text{ }/\text{ }2  \\

=\text{ }\left[ 5~+~8.06 \right]\text{ }/\text{ }2\text{ }or\text{ }\left[ 5~~8.06 \right]\text{ }/\text{ }2  \\

=\text{ }\left[ 13.06 \right]/2\text{ }or\text{ }\left[ -3.06 \right]/2  \\

=\text{ }6.53\text{ }or\text{ }-1.53  \\

\end{array}\]

∴ Value of x = \[6.53\text{ }or\text{ }-1.53\]

(ii) \[{{\mathbf{x}}^{\mathbf{2}}}~+\text{ }\mathbf{7x}\text{ }=\text{ }\mathbf{7}\]

On rearranging the expression, we get

\[{{x}^{2}}~+\text{ }7x\text{ }\text{ }7\text{ }=\text{ }0\]

Let us consider,

\[a\text{ }=\text{ }1,\text{ }b\text{ }=\text{ }7,\text{ }c\text{ }=\text{ }-7\]

So, by using the formula,

\[\begin{array}{*{35}{l}}

D\text{ }=\text{ }{{b}^{2}}~\text{ }4ac  \\

=\text{ }{{\left( 7 \right)}^{2}}~\text{ }4\left( 1 \right)\text{ }\left( -7 \right)  \\

=\text{ }49\text{ }+\text{ }28  \\

=\text{ }77  \\

\end{array}\]

So,

\[\begin{array}{*{35}{l}}

x\text{ }=\text{ }\left[ -7~\pm \text{ }\surd 77 \right]\text{ }/\text{ }2\left( 1 \right)  \\

=\text{ }\left[ -7~\pm ~8.77 \right]\text{ }/\text{ }2  \\

=\text{ }\left[ -7~+~8.77 \right]\text{ }/\text{ }2\text{ }or\text{ }\left[ -7~~8.77 \right]\text{ }/\text{ }2  \\

=\text{ }1.77/2\text{ }or\text{ }-15.77/2  \\

=\text{ }0.885\text{ }or\text{ }-7.885  \\

\end{array}\]

∴ Value of x = \[0.89\text{ }or\text{ }-7.89\]

Solve the following equation and give your answer correct to 3 significant figure:

`5x^2 - 3x - 4 = 0`

Given quadratic equation is `5x^2 - 3x - 4 = 0`

Comparing it with ax2 + bx +c = 0, we get

a = 5, b = -3, c = -4

`:. x = (-b+- sqrt(b^2 - 4ac))/"2a"`

`= (3+-sqrt((-3)^2 - 4 xx 5 xx -4))/(2 xx 5)`

`= (3+-sqrt(9 + 16 xx 5))/10`

`= (3 +-sqrt89)/10`

`= (3 +- 9.433)/10`

`= (3 + 9.433)/10  and (3-9.433)/10`

`= 1.2433 and -0.6433`

= 1.243 and -0.643

Concept: Quadratic Equations

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