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Double brackets and quadratic graphs
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When x is squared, the graph is no longer straight. Quadratics make smooth U-shaped curves, and they come from multiplying two brackets.
Expanding double brackets
Multiply every term in the first bracket by every term in the second.
(x + 3)(x + 5) = x2 + 5x + 3x + 15 = x2 + 8x + 15
A grid helps you keep track.
(x + 3)(x + 5) = x2 + 5x + 3x + 15 = x2 + 8x + 15
A grid helps you keep track.
Quadratic graphs
y = x2 + bx + c makes a smooth U-shaped curve called a parabola. If the x2 term is negative, it is upside down.
Make a table of values, plot the points and join them with a smooth curve, not straight lines.
The curve is symmetrical about a vertical line through its lowest (or highest) point.
Make a table of values, plot the points and join them with a smooth curve, not straight lines.
The curve is symmetrical about a vertical line through its lowest (or highest) point.
Reading the graph
The curve crosses the y-axis at c (when x = 0).
The x-values where it crosses the x-axis (y = 0) are the roots: the solutions of x2 + bx + c = 0.
You can read approximate solutions from any graph this way.
The x-values where it crosses the x-axis (y = 0) are the roots: the solutions of x2 + bx + c = 0.
You can read approximate solutions from any graph this way.
Expand (x − 4)(x + 2).
- x × x = x2, x × 2 = 2x
- −4 × x = −4x, −4 × 2 = −8
- Collect: x2 + 2x − 4x − 8
Answer: x2 − 2x − 8
Expand double brackets by multiplying every term by every term, then collect. Quadratic graphs are U-shaped; their roots are where they cross the x-axis.
The interactive lesson includes the diagrams for this topic.
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