Introduction to Quadratic Equations

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

Advanced25 minLesson
Printable worksheet

Definition

A quadratic equation is a polynomial equation of degree 2. The standard form is:
ax2+bx+c=0ax^2 + bx + c = 0
where:
  • aa, bb, and cc are constants (numbers)
  • a≠0a \neq 0 (if a=0a = 0, it becomes a linear equation)
  • xx is the variable
The graph of a quadratic function f(x)=ax2+bx+cf(x) = ax^2 + bx + c is called a parabola - a U-shaped curve that opens upward when a>0a > 0 or downward when a<0a < 0.

Try it now

Which of the following is a quadratic equation?

Worked Examples

Is 3x2−7x+2=03x^2 - 7x + 2 = 0 a quadratic equation? If so, identify aa, bb, and cc.
1

Check the degree

The highest power of xx is 2 (from 3x23x^2) → Degree = 2
2

Verify a≠0a \neq 0

The coefficient of x2x^2 is 3, which is not zero → a=3≠0a = 3 \neq 0 ✓
3

Identify coefficients

Compare with ax2+bx+c=0ax^2 + bx + c = 0 → a=3a = 3, b=−7b = -7, c=2c = 2

Common Mistakes

Forgetting that aa cannot equal zero

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

Correct: Always check that the coefficient of x2x^2 is non-zero before calling it quadratic.

Sign errors when identifying bb

Why it's wrong: In x2−5x+3=0x^2 - 5x + 3 = 0, students often say b=5b = 5 instead of b=−5b = -5.

Correct: The coefficient includes its sign! Write: x2+(−5)x+3=0x^2 + (-5)x + 3 = 0, so b=−5b = -5.

Confusing the vertex formula

Why it's wrong: Students sometimes use x=b2ax = \frac{b}{2a} instead of x=−b2ax = -\frac{b}{2a}.

Correct: Remember: x=−b2ax = -\frac{b}{2a} (negative sign in front!)

Thinking all parabolas open upward

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

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

Interactive Visual

Quadratic Explorer

y = x² - 4x + 3

Controls width and direction

Shifts the parabola

Y-intercept

xy

Vertex

(2, -1)

Axis of Symmetry

x = 2

Roots (x-intercepts)

x = 3, x = 1

Y-Intercept

(0, 3)

Direction

Opens upward

Discriminant

b² - 4ac = 4

Value Table

x-3-2-10123
y2415830-10
VertexRootsY-InterceptAxis of Symmetry

Interactive Sandbox

Interactive Grapher

-10-50510-10-50510xy

Tap or hover over the graph to read off values.

Try these examples:

y = x^2 - 4x + 3

Expression Calculator

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History

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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=−4.9t2+v0t+h0h = -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=−5t2+20t+1.5h = -5t^2 + 20t + 1.5 meters after tt seconds. The vertex tells us the maximum height.

1Try It Yourself

A basketball player shoots the ball. The height is modeled by h=−4.9t2+9.8t+2h = -4.9t^2 + 9.8t + 2 meters.

When does the ball reach its maximum height?

Step 1: Write the mathematical expression

Use t=−b2at = -\frac{b}{2a} with a=−4.9a = -4.9 and b=9.8b = 9.8:

Business Profit

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

Example:

If profit is P=−2x2+200x−3000P = -2x^2 + 200x - 3000 where xx is the number of items sold, the vertex gives the optimal quantity.

2Try It Yourself

A company's weekly profit is P=−x2+80x−1200P = -x^2 + 80x - 1200 euros, where xx 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 ax2+bx+c=0ax^2 + bx + c = 0 where a≠0a \neq 0
  • 2The graph of a quadratic function is a parabola
  • 3When a>0a > 0, the parabola opens upward (∪); when a<0a < 0, it opens downward (∩)
  • 4The vertex is at x=−b2ax = -\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 (ax2+bx+c=0ax^2 + bx + c = 0) and we solve for specific xx values. A quadratic function is written as f(x)=ax2+bx+cf(x) = ax^2 + bx + c and describes a relationship for all xx values.
A quadratic equation is set equal to zero (ax2+bx+c=0ax^2 + bx + c = 0) and we solve for specific xx values. A quadratic function is written as f(x)=ax2+bx+cf(x) = ax^2 + bx + c and describes a relationship for all xx values.
The word comes from Greek 'parabole' meaning 'comparison' or 'application.' Mathematically, a parabola is defined as all points equidistant from a fixed point (focus) and a fixed line (directrix).
A quadratic equation always has solutions, but they might be complex (imaginary) numbers. If the parabola doesn't cross the x-axis, the solutions are not real numbers.

Glossary

Quadratic equation
A polynomial equation of degree 2 in the form ax2+bx+c=0ax^2 + bx + c = 0 where a≠0a \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=−b2ax = -\frac{b}{2a}
Coefficient
A number multiplied by a variable; in 3x23x^2, the coefficient is 3
Standard form
The arrangement ax2+bx+c=0ax^2 + bx + c = 0 where terms are ordered by decreasing powers of xx
Axis of symmetry
The vertical line x=−b2ax = -\frac{b}{2a} that divides the parabola into two mirror images

Formula Card

Standard Form

ax2+bx+c=0ax^2 + bx + c = 0

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

Vertex x-coordinate

x=−b2ax = -\frac{b}{2a}

Formula to find the x-coordinate of the vertex

Parabola Direction

a>0⇒a > 0 \Rightarrow up, a<0⇒a < 0 \Rightarrow down

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

Today's challengeOne problem, three tries

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