Quadratic Equations: Practice Questions & Notes

A quadratic equation is a polynomial equation of degree 2, which means the highest exponent of the variable is 2.

The standard form of a quadratic equation is:

ax^2 + bx + c = 0

Where:

$a, b, c$ are real numbers
$a \neq 0$
$x$ is the variable

Intuitive Understanding

Think of a quadratic as a U-shaped curve (parabola) on a graph. It represents situations where changes happen non-linearly, like objects thrown into the air or area-related problems.

If linear equations are straight lines, quadratic equations are their curved cousins.

A quadratic equation always has:

At most 2 real roots
Can also have imaginary roots

Roots are the values of $x$ that satisfy the equation — where the parabola intersects the x-axis.

Methods to Solve a Quadratic Equation

Factorization

Used when the equation can be broken down into two binomial terms.

Completing the Square

A universal method, used when factorization doesn’t work easily.

Quadratic Formula

Always applicable.

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

This formula gives:

Two real and distinct roots if $b^2 - 4ac > 0$
One real and equal root if $b^2 - 4ac = 0$
Two imaginary roots if $b^2 - 4ac < 0$

Graphical Method

You draw the parabola and find where it cuts the x-axis (if at all).

Key Terms and Concepts

Term	Explanation
Roots	Values of $x$ that satisfy the quadratic equation
Discriminant	$D = b^2 - 4ac$ , tells nature of roots
Sum of Roots	$-\frac{b}{a}$
Product of Roots	$\frac{c}{a}$
Parabola	Shape of the graph of a quadratic equation
Vertex	The turning point of the parabola, at $x = \frac{-b}{2a}$

Nature of Roots

Discriminant $D = b^2 - 4ac$	Nature of Roots
$D > 0$ and perfect square	Real, distinct, rational
$D > 0$ and not a perfect square	Real, distinct, irrational
$D = 0$	Real, equal
$D < 0$	Complex conjugates

Special Cases to Know

Perfect Square Trinomials

x^2 + 2ax + a^2 = (x + a)^2

Difference of Squares

x^2 - a^2 = (x - a)(x + a)

Zero Product Property
If $ab = 0$ , then either $a = 0$ or $b = 0$
Graph Touches or Cuts X-axis

Touches = One root
Cuts = Two roots
Doesn’t touch = Imaginary roots

Conceptual Tips (Common Mistakes)

Mistake	Tip
Ignoring coefficient of $x^2$	Always divide by $a$ if it’s not 1 before factoring
Wrong sign during splitting middle term	Check that sum and product match with $b$ and $ac$
Misusing ± in quadratic formula	Always include both roots: $\pm$
Assuming all quadratics are factorable	Use discriminant to check first
Not simplifying square roots fully	Always simplify $\sqrt{b^2 - 4ac}$ for accuracy

Word Problems

Quadratic equations show up in:

Motion problems (throwing, catching, speed/time)
Geometry (area, dimensions)
Age problems
Investment/Profit-related problems

Tip: Set up the equation from English by letting unknown = $x$ , then model total using quadratic forms.

Graphical Insight

Equation: $y = ax^2 + bx + c$

If $a > 0$ → Opens upward (U)
If $a < 0$ → Opens downward (∩)
Vertex:

x = \frac{-b}{2a}, \quad y = f\left(\frac{-b}{2a}\right)

Axis of symmetry: $x = \frac{-b}{2a}$

Examples

Example 1: Factorization

Solve: $x^2 + 5x + 6 = 0$

(x + 2)(x + 3) = 0 \Rightarrow x = -2, -3

Example 2: Quadratic Formula

Solve: $2x^2 + 3x - 2 = 0$

a = 2, b = 3, c = -2 \\ D = 9 + 16 = 25 \\ x = \frac{-3 \pm \sqrt{25}}{4} = \frac{-3 \pm 5}{4} \\ x = \frac{2}{4}, \frac{-8}{4} \Rightarrow x = \frac{1}{2}, -2

Example 3: Completing the Square

Solve: $x^2 + 6x + 5 = 0$

x^2 + 6x = -5 \\ x^2 + 6x + 9 = 4 \\ (x + 3)^2 = 4 \Rightarrow x + 3 = \pm 2 \Rightarrow x = -1, -5

Example 4: Word Problem

Q: The product of two consecutive positive integers is 132. Find the integers.
Let the smaller integer be $x$ :

x(x+1) = 132 \Rightarrow x^2 + x - 132 = 0 \Rightarrow (x + 12)(x - 11) = 0 \Rightarrow x = 11