Solve for \( x \) : 1.1.1 \( x^{2}-5 x-50=0 \) \( 1.1 .2 \quad 2 x^{2}-6 x+1=0 \) (correct to TWO decimal places) (3) \( 1.1 .3-x^{2}-3 x>0 \) (3) 1.1.A \( x+3-2 \sqrt{5-x}=0 \) (5) 2 Solve the following equations timulaneously: \[ 2^{x}-2^{r-2}=0 \text { and } x^{x}+2 x y+y^{2}=36 \] A square has sides \( x \) cm each. If each side of the square is increased by 4 cm , the meas is increased by \( 392 \mathrm{~cm}^{2} \). Calculatex \( x \). (4) [24]

To solve the quadratic equations, you can utilize the quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). For the first equation, \( x^2 - 5x - 50 = 0 \), you find the roots by identifying \( a = 1, b = -5, c = -50 \). Plugging these values into the formula gives you the solutions. For \( 1.1.2, 2x^2 - 6x + 1 = 0 \), follow the same process. The roots can be calculated using the quadratic formula and ensure you round to two decimal places as instructed. The inequality \( -x^2 - 3x > 0 \) can be rearranged for easier solving. Factor it to find critical points which can help in determining the intervals that satisfy the inequality. As for the equation \( x + 3 - 2\sqrt{5 - x} = 0 \), isolate the square root and square both sides to eliminate it before solving for \( x \). Now, for simultaneous solutions—consider the implications of the first equation \( 2^x - 2^{r-2} = 0 \); this simplifies to \( 2^x = 2^{r-2} \), leading you to evaluate when \( x = r - 2 \). Then substitute this relationship into the second equation for further simplification. Lastly, the square area problem can be approached algebraically. If the original side length is \( x \), then after the increment the area becomes \( (x+4)^2 \). Set the equation as \( (x + 4)^2 - x^2 = 392 \) to find the value of \( x \). This blend of equations showcases the beauty of algebra in solving real-world problems—from managing spaces in geometry to harnessing number properties in quadratic equations! Always double-check your calculations to avoid common pitfalls like misplacing negative signs or forgetting to include all possible roots.