Let’s take a look at this A maths question on **Equations and Inequalities.** A handful of students will be confused with the keywords or how to effectively determine the correct sign of the discriminant.

The example below is very similar to **St Margaret 2016** A Math Prelim Paper 1 Q2.

Original Question: Find the range of values of \(p\) for which \((p+2)x^2 – 12x + 2(p-1)\) is always negative.

__Modified Example__: Find the range of values of \(p\) for which \((p+2)x^2 – 12x > -2(p-1)\) for all real values of \(x\).

**Misconception**:

Students will mistaken REAL as the keyword and likely think that \(b^2-4ac \geq 0\).

The emphasis is that the word REAL is crucial ONLY if coupled with a friend called ROOTS. We need to see the Keyword – **REAL ROOTS** for \(b^2-4ac \geq 0\) to hold true.

**Step 1:**

Ensure that the right hand side of the inequality is** zero**.

**Step 2:**

Having done so, recognise the keyword as “\(>0\)“. This implies that the graph will be above the axis and the shape of this quadratic expression will be a **Happy Face**. Since the U-shape is **not intersecting** with the axis, the solutions will be **imaginary** and \(b^2-4ac < 0\).

**Keyword **\(>0\) implies

\(\Rightarrow\) Happy Face is ** ABOVE** axis

\(\Rightarrow\) ** No** Real Solutions

\(\Rightarrow\) \(b^2-4ac < 0\)

**Step 3:**

Observe \(a=p+2, b=-12\) and \(c=2p-2\)

For imaginary solutions, we have :

\(b^2-4ac<0\)

\((-12)^2-4(p+2)(2p-2)<0\)

\(144-4(2p^2-2p+4p-4)<0\)

\(-8p^2-8p+160<0\)

\(-p^2-p+20<0\)

\((-p+4)(p+5)<0\)

\(p<-5\) or \( p>4\)…..(1)

At this stage, most students are contented with getting the solutions and will think that the question is completed.

Well, we are almost there. This is a **typical example** of having a **variable coefficient** (\(p+2\)) in front of \(x^2\).

__Extra Step is mandatory__

Remember we mentioned earlier that the graph is a Happy Face, thus we expect the coefficient of \(x^2\) to be positive and we have

\(p+2>0\)

\(p>-2\)….. (2)

Visualise the overlapping region of answers on a **number line **from (1) and (2),

Therefore \(p>4\) (final solution)