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QUESTION 1 Solve for \( x \) : 1.1.1 \( \quad x^{2}-7 x+12=0 \) \( 1.1 .2 \quad x(3 x+5)=1 \quad \) (correct to TWO decimal pla \( 1.1 .3 \quad x^{2}<-2 x+15 \) \( 1.1 .4 \quad \sqrt{2(1-x)}=x-1 \) \( 1.2 \quad \) Solve for \( x \) and \( y \) simultaneously: \( 3^{x+y}=27 \quad \) and \( \quad x^{2}+y^{2}=17 \)

Ask by Daniel Dickson. in South Africa
Mar 15,2025

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**1.1.1** \( x = 3 \) or \( x = 4 \) **1.1.2** \( x \approx 0.18 \) or \( x \approx -1.85 \) **1.1.3** \( -5 < x < 3 \) **1.1.4** \( x = 1 \) **1.2** \( (x, y) = (4, -1) \) or \( (x, y) = (-1, 4) \)

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Beyond the Answer

To deliver some extra insight on solving quadratic equations and working with simultaneous equations, let’s dive into a bit of mathematical history and some practical tips! Did you know that the quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), has been around since ancient Babylon? They had developed methods to solve quadratic equations as far back as 2000 BC, using geometric interpretations and algorithms that resemble our formula today. It’s a testament to how long humanity has been fascinated by finding the roots of equations! When tackling simultaneous equations, especially like the ones given, a common mistake is to substitute values without checking their validity first. Always organize your work and carefully keep track of each equation. A nifty tip is to express one variable in terms of another from one equation before substituting it into the other (like from \(3^{x+y}=27\), you can quickly see \(x+y=3\)). This clear pathway often leads you to the correct solutions more efficiently!

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