Solving Quadratic Equations: Graphical Methods

Drawing Quadratic Graphs

Connecting plotted points with a ruler might seem like the neatest way to draw a graph, but for quadratics, doing this will immediately cost you marks. All quadratic functions of the form $y = ax^2 + bx + c$ must be drawn freehand as a smooth, continuous curve called a parabola.

If the value of $a$ is positive ( $a > 0$ ), the parabola will be a U-shaped curve with a minimum turning point (vertex). If $a$ is negative ( $a < 0$ ), the curve will be n-shaped with a maximum turning point. Parabolic graphs are perfectly symmetrical, with a vertical line of symmetry ( $x = k$ ) passing exactly through the turning point.

What are Roots?

The roots of a quadratic equation $ax^2 + bx + c = 0$ are the $x$ -values where the function equals zero. On a graph, these are the $x$ -intercepts, which are the points where the curve crosses the $x$ -axis (the line $y = 0$ ).

A quadratic graph can have two solutions if it crosses the axis twice, or exactly one solution if it just touches the axis at its turning point (a repeated root). It will have zero solutions if the curve does not cross or touch the $x$ -axis at all. In Edexcel exams, values read from a graph are known as an approximate solution because the exact value depends on the accuracy of your hand-drawn curve and the scale of the axes.

Worked Example: Finding Roots

Solve $x^2 - x - 6 = 0$ graphically for $-3 \leq x \leq 4$ .

Step 1: Create a table of values by substituting the $x$ -values into $y = x^2 - x - 6$ .

$x = -3: y = (-3)^2 - (-3) - 6 = 9 + 3 - 6 = 6$
$x = -2: y = (-2)^2 - (-2) - 6 = 0$
$x = -1: y = (-1)^2 - (-1) - 6 = -4$
$x = 0: y = -6$
$x = 1: y = (1)^2 - (1) - 6 = -6$
$x = 2: y = (2)^2 - (2) - 6 = -4$
$x = 3: y = (3)^2 - (3) - 6 = 0$
$x = 4: y = (4)^2 - (4) - 6 = 6$

Step 2: Plot the points on a coordinate grid.

Mark the coordinates $(-3, 6), (-2, 0), (-1, -4), (0, -6), (1, -6), (2, -4), (3, 0), (4, 6)$ on the grid.
Points must be plotted within a strict tolerance of $\pm 1$ small square.

Step 3: Draw the curve.

Connect the points with a single, smooth freehand curve.
Ensure the base between $x = 0$ and $x = 1$ is a rounded U-shape and not a flat, horizontal line.

Step 4: Identify the roots where the curve intersects the $x$ -axis.

The curve crosses the $x$ -axis ( $y = 0$ ) at the points $(-2, 0)$ and $(3, 0)$ .

Step 5: State the final solutions.

$x = -2$ and $x = 3$

Solving Equations with a Straight Line

Sometimes you will be asked to solve a related equation like $ax^2 + bx + c = k$ using a pre-drawn graph. To do this, you must draw a suitable straight line horizontally at $y = k$ across the grid. The solutions are the $x$ -coordinates of the intersection point where the parabola and the new straight line meet.

If the equation does not perfectly match the graph you have drawn, you must rearrange it first to isolate the original function. For example, to solve $x^2 - 5x + 3 = 0$ using a pre-drawn graph of $y = x^2 - 5x$ , you must rearrange the equation to $x^2 - 5x = -3$ and then draw the line $y = -3$ .

Worked Example: Intersection with a Line

Solve $x^2 - x - 6 = -2$ graphically.

Step 1: Plot the curve for the original quadratic function.

Draw the graph of $y = x^2 - x - 6$ (following the plotting steps from the previous example).

Step 2: Draw the suitable straight line.

Draw the horizontal line $y = -2$ directly across the coordinate grid.

Step 3: Find the intersections and read the $x$ -values.

Mark the two points where the parabola crosses the line $y = -2$ .
Project vertically down from these specific points to the $x$ -axis.

Step 4: State the final approximate solutions.

$x = -1.6$ and $x = 2.6$

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Exam Tips

Common Mistake
If asked to write down the 'roots' or 'solutions', students often write down full coordinates (e.g., $(-2, 0)$ ). You must only write the $x$ -values.
2
If an exam question states 'Use the graph to find...', you MUST read your answers directly from the graph. Using the quadratic formula algebraically will score zero marks, even if your answer is mathematically perfect.
3
Always check the axis scale before reading your approximate solutions. If 10 small squares equal 1 unit, each square is 0.1, but if 5 squares equal 1 unit, each square is 0.2.
4
Use the TABLE function on your scientific calculator to generate $y$ -values quickly and avoid arithmetic errors. Remember to put negative $x$ -values inside brackets when squaring them, like $(-3)^2$ .

Key Terms(8)

Parabola: The name of the symmetrical U-shaped or n-shaped curve formed by a quadratic function.
Turning point (vertex): The maximum or minimum point of the parabola where the curve changes direction.
Roots: The $x$ -values for which a mathematical function equals zero; graphically, these are the $x$ -intercepts.
x-intercept: The point where a graph crosses the $x$ -axis, meaning the $y$ -coordinate is always exactly $0$ .
Approximate solution: A value read from a graph (typically required to 1 decimal place) that is close to the true algebraic solution.
Table of values: A grid used to calculate $y$ -coordinates for specific $x$ -inputs before plotting them on axes.
Suitable straight line: An auxiliary line (either $y = k$ or $y = mx + c$ ) drawn on a grid to help solve a quadratic equation graphically.
Intersection point: The coordinate $(x, y)$ where two separate graphs meet on a grid.

Previous NoteSolving Quadratic Equations: The Quadratic Formula Next Subtopic: Simultaneous EquationsSimultaneous Equations

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Key Terms(8)

Parabola: The name of the symmetrical U-shaped or n-shaped curve formed by a quadratic function.
Turning point (vertex): The maximum or minimum point of the parabola where the curve changes direction.
Roots: The $x$ -values for which a mathematical function equals zero; graphically, these are the $x$ -intercepts.
x-intercept: The point where a graph crosses the $x$ -axis, meaning the $y$ -coordinate is always exactly $0$ .
Approximate solution: A value read from a graph (typically required to 1 decimal place) that is close to the true algebraic solution.
Table of values: A grid used to calculate $y$ -coordinates for specific $x$ -inputs before plotting them on axes.
Suitable straight line: An auxiliary line (either $y = k$ or $y = mx + c$ ) drawn on a grid to help solve a quadratic equation graphically.
Intersection point: The coordinate $(x, y)$ where two separate graphs meet on a grid.

Solving Quadratic Equations: Graphical Methods

Drawing Quadratic Graphs

What are Roots?

Worked Example: Finding Roots

Solve $x^2 - x - 6 = 0$ graphically for $-3 \leq x \leq 4$ .

Step 1: Create a table of values by substituting the $x$ -values into $y = x^2 - x - 6$ .

$x = -3: y = (-3)^2 - (-3) - 6 = 9 + 3 - 6 = 6$
$x = -2: y = (-2)^2 - (-2) - 6 = 0$
$x = -1: y = (-1)^2 - (-1) - 6 = -4$
$x = 0: y = -6$
$x = 1: y = (1)^2 - (1) - 6 = -6$
$x = 2: y = (2)^2 - (2) - 6 = -4$
$x = 3: y = (3)^2 - (3) - 6 = 0$
$x = 4: y = (4)^2 - (4) - 6 = 6$

Step 2: Plot the points on a coordinate grid.

Mark the coordinates $(-3, 6), (-2, 0), (-1, -4), (0, -6), (1, -6), (2, -4), (3, 0), (4, 6)$ on the grid.
Points must be plotted within a strict tolerance of $\pm 1$ small square.

Step 3: Draw the curve.

Connect the points with a single, smooth freehand curve.
Ensure the base between $x = 0$ and $x = 1$ is a rounded U-shape and not a flat, horizontal line.

Step 4: Identify the roots where the curve intersects the $x$ -axis.

The curve crosses the $x$ -axis ( $y = 0$ ) at the points $(-2, 0)$ and $(3, 0)$ .

Step 5: State the final solutions.

$x = -2$ and $x = 3$

Solving Equations with a Straight Line

Worked Example: Intersection with a Line

Solve $x^2 - x - 6 = -2$ graphically.

Step 1: Plot the curve for the original quadratic function.

Draw the graph of $y = x^2 - x - 6$ (following the plotting steps from the previous example).

Step 2: Draw the suitable straight line.

Draw the horizontal line $y = -2$ directly across the coordinate grid.

Step 3: Find the intersections and read the $x$ -values.

Mark the two points where the parabola crosses the line $y = -2$ .
Project vertically down from these specific points to the $x$ -axis.

Step 4: State the final approximate solutions.

$x = -1.6$ and $x = 2.6$

Sign up to continue reading

Get full access to revision notes, key terms, and exam tips for every subtopic.

Exam Tips

Common Mistake
If asked to write down the 'roots' or 'solutions', students often write down full coordinates (e.g., $(-2, 0)$ ). You must only write the $x$ -values.
2
If an exam question states 'Use the graph to find...', you MUST read your answers directly from the graph. Using the quadratic formula algebraically will score zero marks, even if your answer is mathematically perfect.
3
Always check the axis scale before reading your approximate solutions. If 10 small squares equal 1 unit, each square is 0.1, but if 5 squares equal 1 unit, each square is 0.2.
4
Use the TABLE function on your scientific calculator to generate $y$ -values quickly and avoid arithmetic errors. Remember to put negative $x$ -values inside brackets when squaring them, like $(-3)^2$ .

Key Terms(8)

Parabola: The name of the symmetrical U-shaped or n-shaped curve formed by a quadratic function.
Turning point (vertex): The maximum or minimum point of the parabola where the curve changes direction.
Roots: The $x$ -values for which a mathematical function equals zero; graphically, these are the $x$ -intercepts.
x-intercept: The point where a graph crosses the $x$ -axis, meaning the $y$ -coordinate is always exactly $0$ .
Approximate solution: A value read from a graph (typically required to 1 decimal place) that is close to the true algebraic solution.
Table of values: A grid used to calculate $y$ -coordinates for specific $x$ -inputs before plotting them on axes.
Suitable straight line: An auxiliary line (either $y = k$ or $y = mx + c$ ) drawn on a grid to help solve a quadratic equation graphically.
Intersection point: The coordinate $(x, y)$ where two separate graphs meet on a grid.

Previous NoteSolving Quadratic Equations: The Quadratic Formula Next Subtopic: Simultaneous EquationsSimultaneous Equations

Back to Quadratic Equations

Put your knowledge into practice — try past paper questions for Mathematics

Key Terms(8)

Parabola: The name of the symmetrical U-shaped or n-shaped curve formed by a quadratic function.
Turning point (vertex): The maximum or minimum point of the parabola where the curve changes direction.
Roots: The $x$ -values for which a mathematical function equals zero; graphically, these are the $x$ -intercepts.
x-intercept: The point where a graph crosses the $x$ -axis, meaning the $y$ -coordinate is always exactly $0$ .
Approximate solution: A value read from a graph (typically required to 1 decimal place) that is close to the true algebraic solution.
Table of values: A grid used to calculate $y$ -coordinates for specific $x$ -inputs before plotting them on axes.
Suitable straight line: An auxiliary line (either $y = k$ or $y = mx + c$ ) drawn on a grid to help solve a quadratic equation graphically.
Intersection point: The coordinate $(x, y)$ where two separate graphs meet on a grid.