What is a Quadratic Equation?

Quadratic Equation

A quadratic equation is any equation that can be written in the standard form \(ax^2 + bx + c = 0\) where \(a \neq 0\).

It is called “quadratic” because the highest power of the variable is 2.

Graph of a Quadratic Function

Parabola for y = x² Parabola for y = x² A smooth curve opening upwards. x y

A smooth curve opening upwards.

What shape do we get?

Plotting \(y = x^{2}\) joins the points into a smooth U-shape.

This U-shaped curve is called a parabola. Every quadratic graph \(y = ax^{2}+bx+c\) forms a parabola.

Key Points:

  • U-shaped curve is called a parabola.
  • All graphs of \(y = ax^{2} + bx + c\) are parabolas.

Roots as X-Intercepts

Graph of y = x² − 6x + 8 y = x² − 6x + 8 X-intercepts at x = 2 and x = 4 represent the roots. 2 4 y x

X-intercepts at x = 2 and x = 4 represent the roots.

Graphical Meaning of Roots

A root is any x-value where the parabola meets the x-axis, so \(y = 0\).

On this graph, the quadratic \(y = x^{2} - 6x + 8\) touches the axis at \(x = 2\) and \(x = 4\).

Key Points:

  • Roots are the solutions of the quadratic equation.
  • They appear graphically where the curve crosses the x-axis.

Procedure to Solve by Graphing

Follow these steps to obtain the roots of a quadratic equation from its graph.

1

Prepare a value table

Select several x-values and calculate corresponding y = ax² + bx + c.

2

Plot the computed points

Mark each (x, y) accurately on graph paper with equal units.

3

Draw the parabola

Connect the points smoothly to form the quadratic curve.

4

Locate the x-intercepts

Identify where the curve intersects the x-axis.

5

Write the roots

The x-coordinates of the intercepts are the equation’s solutions.

Pro Tip:

Choose x-values symmetrically around the vertex for a balanced, accurate graph.

Worked Example: x² − 5x + 6 = 0

y = x² − 5x + 6

Graph of y = x² − 5x + 6 Roots appear where the curve meets the x-axis at x = 2 and x = 3. x y

Roots at x = 2 and x = 3.

Key Insights

1. Plot ordered pairs for \(x = 0\) to \(5\).

2. Join points smoothly to form the parabola.

3. Curve intersects the x-axis at \(x = 2\) and \(x = 3\).

4. Therefore, the roots of the quadratic are 2 and 3.

Legend

Parabola \(y = x^{2}-5x+6\)
Roots

Check Your Understanding

Question

From the graph shown, what are the roots of the quadratic equation? Identify the x-intercepts.

1
1 and 5
2
2 and 3
3
3 and 4
4
0 and 6

Hint:

Roots are the x-coordinates where the curve meets the x-axis (y = 0).

Special Root Scenarios

root_scenarios.png

Graphical view of root scenarios

Reading the x-axis

Count the parabola's hits on the x-axis to determine its real roots.

This visual cue separates two, one, and zero-root cases at a glance.

Key Points:

  • Two roots – curve crosses the x-axis twice.
  • One root – curve touches the x-axis only at its vertex.
  • No real roots – curve stays entirely above or below the x-axis.

Key Takeaways

Every quadratic equation plots as a U-shaped parabola.

Roots are the x-intercepts where \(y = 0\).

Plot a few precise points to reveal those intercepts.

The parabola’s position shows whether there are two, one, or no real solutions.

Thank You!

These points consolidate the graphical solution method.