Simultaneous Linear and Quadratic Equations - SS2 Mathematics Past Questions and Answers - page 1

\[x^{2} + 5x + 4 = 0\]

\[x^{2} + 5x = - 4\]

\[x^{2} + 5x + \left( \frac{5}{2} \right)^{2} = \left( \frac{5}{2} \right)^{2} - 4\]

\[x^{2} + 5x + \frac{25}{4} = \frac{25}{4} - 4\]

\[{(x + \frac{5}{2})}^{2} = \frac{9}{4}\]

\[x + \frac{5}{2} = \pm \sqrt{\frac{9}{4}} = \pm \frac{3}{2}\]

\[x + \frac{5}{2} = \pm \frac{3}{2}\]

\[x = \frac{3}{2} - \frac{5}{2}\ or - \frac{3}{2} - \frac{5}{2}\]

\[x = \frac{3 - 5}{2}\ or\frac{- 3 - 5}{2}\]

\[x = \frac{- 2}{2}\ or\frac{- 8}{2}\]

\(x = - 1\ or - 4\)

\(x + y = 8\) (1)

\(x^{2} - y^{2} = 16\) (2)

From (1), transpose \(y\)

\[x + y = 8\]

\[y = 8 - x\]

Substitute the value of \(y\) in (2)

\[x^{2} - {(8 - x)}^{2} = 16\]

\[16x - 64 = 16\]

\[16x = 16 + 64\]

\[16x = 80\]

\[16x - 80 = 0\]

\[16(x - 5) = 0\]

\[\therefore x = 5\]

Substitute the values of \(x\) in (1)

\[x + y = 8\]

\[5 + y = 8\]

\[y = 8 - 5 = 3\]

The solution to the simultaneous equations \(2x + y = 5\) and \(x^{2} + y^{2} = 25\) is the ordered pairs: \((5;3)\)

\[{2x}^{2} + 9x = 5\]

\[x^{2} + \frac{9}{2}x = \frac{5}{2}\]

\[x^{2} + \frac{9}{2}x + {(\frac{9}{4})}^{2} = \frac{5}{2} + {(\frac{9}{4})}^{2}\]

\[x^{2} + \frac{9}{2}x + \frac{81}{16} = \frac{5}{2} + \frac{81}{16}\]

\[{(x + \frac{9}{4})}^{2} = \frac{5}{2} + \frac{81}{16}\]

\[{(x + \frac{9}{4})}^{2} = \frac{40 + 81}{16}\]

\[{(x + \frac{9}{4})}^{2} = \frac{121}{16}\]

\[x + \frac{9}{4} = \pm \sqrt{\frac{121}{16}} = \pm \frac{11}{4}\]

\[x = - \frac{9}{4} \pm \frac{11}{4}\]

\[x = - \frac{9}{4} + \frac{11}{4}\ \ \ \ \ or\ \ \ \ \ - \frac{9}{4} - \frac{11}{4}\]

\(x = \frac{11 - 9}{4}\ \ \ \ or\ \frac{- 9 - 11}{4}\)

\(x = \frac{1}{2}\) or \(- 5\)

1. **Express y in terms of x from the linear equation:** From \(3x + y = 10\), we get \(y = 10 - 3x\). 2. **Substitute y into the quadratic equation:** Substitute \(y = 10 - 3x\) into \(2x^2 + y^2 = 19\): \(2x^2 + (10 - 3x)^2 = 19\) 3. **Expand and simplify the quadratic equation:** \(2x^2 + (100 - 60x + 9x^2) = 19\) \(11x^2 - 60x + 100 =