Quadratic Equations

Quadratics are equations in the form ax²+bx+c=0 where a, b, and c are constants. They can be solved by factorising into the form x-m$$x-n=0, which gives us solutions x=m and x=n.

When factorising isn't straightforward, we can use the quadratic formula: x = $-b±√(b²-4ac)$/2a. This formula works for any quadratic equation regardless of complexity.

With inequalities involving quadratics, remember that the graph's shape affects the solution. When a is positive, the parabola opens upward (creating a minimum); when a is negative, it opens downward (creating a maximum).

Quick Tip: When solving quadratic inequalities, always sketch the curve to visualise where the function is positive (above x-axis) or negative (below x-axis).

Set Notation and The Discriminant

Set notation provides a concise way to describe intervals of values. The notation {x : P} represents all values of x that satisfy condition P. For intervals, we use parentheses () for exclusive bounds and square brackets [] for inclusive bounds.

For example, x < m or x > n can be written as $-∞, m$ ∪ (n, ∞), while s ≤ x < t would be [s, t).

The discriminant $b²-4ac$ tells us about the nature of a quadratic equation's solutions:

• If b²-4ac > 0: two distinct real roots
• If b²-4ac = 0: one repeated real root
• If b²-4ac < 0: no real roots (only complex solutions)

Remember: The discriminant is your quick diagnostic tool - it reveals everything about a quadratic's solutions without requiring you to solve the equation fully!

Understanding Quadratic Roots

When solving quadratics, the discriminant $b²-4ac$ helps determine the number and type of solutions:

A positive discriminant $b²-4ac > 0$ gives two different real roots. For example, x²-4x+3=0 has solutions x=1 and x=3.

A zero discriminant $b²-4ac = 0$ produces one repeated root, often called a double root. The equation can be written in the form $x-k$². For example, x²-4x+4=0 gives us $x-2$² = 0, so x=2.

A negative discriminant $b²-4ac < 0$ means there are no real roots. The quadratic never crosses the x-axis.

Exam Tip: Questions often ask you to find values of parameters that give specific types of roots. Always use the discriminant conditions to solve these problems!

Solving Techniques

Completing the square transforms a quadratic into the form a$(x+b/2a)²-(b/2a)²+c/a$. This technique helps find the vertex of parabolas and sometimes simplifies solving.

For simultaneous equations, we have two main approaches:

Method 1 (Elimination): Manipulate equations to eliminate one variable. For example, with 3x+y=29 and 4x+3y=47, multiply the first equation by 3 to get 9x+3y=87. Subtracting the second equation eliminates y, giving 5x=40, so x=8. Substitute back to find y=5.

Method 2 (Substitution): Rearrange one equation to express one variable in terms of another, then substitute. From 3x+y=29, we get y=29-3x. Substituting into 4x+3y=47 leads to 4x+3$29-3x$=47, which simplifies to x=8 and y=5.

Challenge yourself: Try both methods on the same problem to see which one feels more intuitive for you!

Solving Non-Linear Simultaneous Equations

Non-linear simultaneous equations involve at least one equation that isn't a straight line. These typically represent the points where two different curves intersect.

For these problems, substitution (Method 2) is almost always the best approach. For example, to find where y=x²-4x+3 and y=3-x intersect:

1. Set the expressions equal: x²-4x+3 = 3-x
2. Rearrange to standard form: x²-3x+0=0
3. Factorise: x$x-3$=0, giving x=0 or x=3
4. Substitute each x-value back to find corresponding y-values

The solution points are (0,3) and (3,0), representing the exact coordinates where these two curves meet.

Visual insight: Whenever you solve these problems, sketch the curves to verify your answers make sense geometrically!

Quadratic Functions and Their Graphs

The shape of a quadratic function y=ax²+bx+c depends critically on the value of a:

When a is positive, the parabola opens upward, creating a minimum point. This means the function has its lowest value at the vertex, and increases as x moves away in either direction.

When a is negative, the parabola opens downward, creating a maximum point. Here, the function reaches its highest value at the vertex.

The vertex form y=a$x-h$²+k directly gives the turning point (h,k), making it particularly useful for identifying these critical points on the graph.

Practical application: Many optimization problems in real life use quadratics - finding minimum costs or maximum profits often involves finding the vertex of a quadratic function!