Quadratics: solution of quadratic equations, nature of roots, quadratic inequalities

Quadratics: Solution of Quadratic Equations, Nature of Roots, Quadratic Inequalities

1. Solving Quadratic Equations

Quadratic equations have the form $ax^2+bx+c=0$ where $aeq0$. There are three common methods to find the solutions (roots).

1. Factorisation – write the quadratic as a product of two binomials. Example: $x^2-5x+6=0$ can be written as $(x-2)(x-3)=0$, giving $x=2$ or $x=3$.
2. Quadratic Formula – $$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$ The term under the square root, $b^2-4ac$, is called the discriminant.
3. Completing the Square – rewrite $ax^2+bx+c$ as $a(x-h)^2+k$ and solve for $x$. This method is useful when the quadratic is not easily factorable.

🔍 Tip: Always check if the quadratic can be factored before using the formula – it saves time on the exam.

2. Nature of Roots

The discriminant $D=b^2-4ac$ tells us about the roots:

3. Quadratic Inequalities

To solve inequalities such as $ax^2+bx+c \, \mathcal{R} \, 0$ (where $\mathcal{R}$ is $>$, $<$, $\ge$, or $\le$), follow these steps:

1. Find the roots of the associated equation $ax^2+bx+c=0$.
2. Plot the roots on a number line and divide it into intervals.
3. Choose a test point from each interval and evaluate the sign of $ax^2+bx+c$.
4. Use the sign to determine which intervals satisfy the inequality.

🧩 Example: Solve $x^2-4x-5>0$.

• Roots: $x=5$ and $x=-1$ (factorise: $(x-5)(x+1)>0$).
• Intervals: $(-\infty,-1)$, $(-1,5)$, $(5,\infty)$.
• Test points: $-2$, $0$, $6$ → signs: $+$, $-$, $+$.
• Solution: $x\in(-\infty,-1)\cup(5,\infty)$.