Introduction to Quadratic Equations

Learn what quadratic equations are, their standard form, and how to identify their key features.

Advanced25 minLesson

Definition

A quadratic equation is a polynomial equation of degree 2. The standard form is:

ax^2 + bx + c = 0

where:

$a$ , $b$ , and $c$ are constants (numbers)
$a \neq 0$ (if $a = 0$ , it becomes a linear equation)
$x$ is the variable

The graph of a quadratic function

f(x) = ax^2 + bx + c

is called a parabola - a U-shaped curve that opens upward when

a > 0

or downward when

a < 0

Try it now

Which of the following is a quadratic equation?

Worked Examples

Is $3x^2 - 7x + 2 = 0$ a quadratic equation? If so, identify $a$ , $b$ , and $c$ .

Check the degree

The highest power of $x$ is 2 (from $3x^2$ ) → Degree = 2

Verify $a \neq 0$

The coefficient of $x^2$ is 3, which is not zero → $a = 3 \neq 0$ ✓

Identify coefficients

Compare with $ax^2 + bx + c = 0$ → $a = 3$ , $b = -7$ , $c = 2$

Answer: Yes, it is a quadratic equation with $a = 3$ , $b = -7$ , and $c = 2$ .

Common Mistakes

Forgetting that $a$ cannot equal zero

Why it's wrong: If $a = 0$ , the equation becomes $bx + c = 0$ , which is linear, not quadratic.

Correct: Always check that the coefficient of $x^2$ is non-zero before calling it quadratic.

Sign errors when identifying $b$

Why it's wrong: In $x^2 - 5x + 3 = 0$ , students often say $b = 5$ instead of $b = -5$ .

Correct: The coefficient includes its sign! Write: $x^2 + (-5)x + 3 = 0$ , so $b = -5$ .

Confusing the vertex formula

Why it's wrong: Students sometimes use $x = \frac{b}{2a}$ instead of $x = -\frac{b}{2a}$ .

Correct: Remember: $x = -\frac{b}{2a}$ (negative sign in front!)

Thinking all parabolas open upward

Why it's wrong: The direction depends on the sign of $a$ , not on any other coefficient.

Correct: If $a > 0$ : opens upward (∪). If $a < 0$ : opens downward (∩).

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Practice Problems

15 problems

Problem 1 of 15

Easy

Which of the following is a quadratic equation?

Why It Matters

Quadratic equations appear everywhere in the real world:

Physics: The path of a thrown ball follows a parabola. The equation $h = -4.9t^2 + v_0t + h_0$ models projectile motion.
Architecture: Parabolic arches are used in bridges and buildings because they distribute weight efficiently.
Business: Profit functions are often quadratic - there's an optimal price that maximizes revenue.
Sports: The trajectory of a basketball shot, a soccer kick, or a golf swing all follow quadratic paths.

Understanding quadratics helps us model and predict real-world behavior!

Real World Applications

Projectile Motion

When you throw a ball, its height over time follows a quadratic equation.

Example:

A ball thrown upward has height $h = -5t^2 + 20t + 1.5$ meters after $t$ seconds. The vertex tells us the maximum height.

1Try It Yourself

A basketball player shoots the ball. The height is modeled by $h = -4.9t^2 + 9.8t + 2$ meters.

When does the ball reach its maximum height?

Step 1: Write the mathematical expression

Use $t = -\frac{b}{2a}$ with $a = -4.9$ and $b = 9.8$ :

Business Profit

Companies use quadratic models to find the price that maximizes profit.

Example:

If profit is $P = -2x^2 + 200x - 3000$ where $x$ is the number of items sold, the vertex gives the optimal quantity.

2Try It Yourself

A company's weekly profit is $P = -x^2 + 80x - 1200$ euros, where $x$ is units sold.

How many units should they sell to maximize profit?

Step 1: Write the mathematical expression

Find the x-coordinate of the vertex:

Key Takeaways

1A quadratic equation has the form $ax^2 + bx + c = 0$ where $a \neq 0$
2The graph of a quadratic function is a parabola
3When $a > 0$ , the parabola opens upward (∪); when $a < 0$ , it opens downward (∩)
4The vertex is at $x = -\frac{b}{2a}$ , and represents the minimum or maximum point
5Quadratic equations model real-world situations like projectile motion and profit optimization

Frequently Asked Questions

A quadratic equation is set equal to zero (

ax^2 + bx + c = 0

) and we solve for specific

x

values. A quadratic function is written as

f(x) = ax^2 + bx + c

and describes a relationship for all

x

values.

Glossary

Quadratic equation: A polynomial equation of degree 2 in the form $ax^2 + bx + c = 0$ where $a \neq 0$
Parabola: The U-shaped curve that is the graph of a quadratic function
Vertex: The highest or lowest point on a parabola; occurs at $x = -\frac{b}{2a}$
Coefficient: A number multiplied by a variable; in $3x^2$ , the coefficient is 3
Standard form: The arrangement $ax^2 + bx + c = 0$ where terms are ordered by decreasing powers of $x$
Axis of symmetry: The vertical line $x = -\frac{b}{2a}$ that divides the parabola into two mirror images

Formula Card

Standard Form

ax^2 + bx + c = 0

The standard form of a quadratic equation where $a \neq 0$

Vertex x-coordinate

x = -\frac{b}{2a}

Formula to find the x-coordinate of the vertex

Parabola Direction

a > 0 \Rightarrow

up,

a < 0 \Rightarrow

down

The sign of $a$ determines if the parabola opens upward or downward