Algebra and Functions
Quadratics
Pearson Edexcel International Advanced Level Mathematics
Completing the square
- Completed square form: a(x + b)2 + c, or a + b(x + c)2 when the question gives that form.
- x2 + px = (x + p2)2 − (p2)2
- When the coefficient of x2 is not 1, take it out as a factor of the x2 and x terms first.
- Take the coefficient of x2 out of the x2 and x terms.
- Inside the bracket, halve the coefficient of x: write the squared bracket and subtract the square of that half.
- Multiply out the outer factor and collect the constants.
Turning point and sketch
- For y = a(x + b)2 + c, the turning point is (−b, c) and the line of symmetry is x = −b.
- A minimum point when a > 0; a maximum point when a < 0.
- A sketch shows the shape, the turning point, and the coordinates where the curve meets the axes.
f(x) = 7 + 12x − 3x2
(a) Express f(x) in the form a + b(x + c)2, where a, b and c are integers.
(b) Hence write down the coordinates of the maximum point of the curve with equation y = f(x).
(a) −3(x2 − 4x) + 7
−3[(x − 2)2 − 4] + 7
−3(x − 2)2 + 12 + 7
19 − 3(x − 2)2
(b) (2, 19)
The discriminant
For ax2 + bx + c = 0, the quadratic formula is x = −b ± √b2 − 4ac2a and the discriminant is b² − 4ac.
| Discriminant | Roots |
|---|---|
| b² − 4ac > 0 | two distinct real roots |
| b² − 4ac = 0 | equal roots (one repeated root) |
| b² − 4ac < 0 | no real roots |
- A line touches a curve (is a tangent to it) when substituting gives a quadratic with b² − 4ac = 0.
- For equal roots, set b² − 4ac = 0 and solve the equation. Never write an inequality for the equal roots condition.
- Real roots (equal or distinct): b² − 4ac ≥ 0.
- Rearrange the equation to the form ax2 + bx + c = 0.
- Write a, b and c in terms of k.
- Write b² − 4ac in terms of k and apply the condition: > 0, = 0 or < 0.
- Solve the quadratic inequality in k: find the critical values, then sketch to choose the inside or outside region.
- Where k is the coefficient of x2, check whether k = 0 must be excluded.
When the critical values of k are surds, never write decimal values of k. A decimal is not an exact value. Write the exact surd form, such as p ± q√r, and keep it exact in any inequality that follows.
The equation kx2 + 4x + k = 3, where k is a constant, has no real roots.
Find the range of possible values of k.
kx2 + 4x + (k − 3) = 0
b² − 4ac = 42 − 4k(k − 3) < 0
16 − 4k2 + 12k < 0 ⇒ k2 − 3k − 4 > 0
(k − 4)(k + 1) > 0, critical values k = −1, 4
k < −1 or k > 4
Quadratics in a function of x
- An equation in a2x and ax is a quadratic in p = ax, since a2x = p2.
- Rewrite each term with the index laws: ax + n = anp and a2x + n = anp2.
- The same idea applies to a quadratic in x2, x3 or √x.
- Solve for p, then set ax equal to each value and solve for x.
In this question you must show all stages of your working.
(a) Given that p = 3x, show that the equation 32x + 1 + 3 = 3x + 2 + 3x can be written as 3p2 − 10p + 3 = 0
(b) Hence solve 32x + 1 + 3 = 3x + 2 + 3x
(a) 32x + 1 = 3(3x)2 = 3p2 and 3x + 2 = 9 × 3x = 9p
3p2 + 3 = 9p + p
3p2 − 10p + 3 = 0
(b) (3p − 1)(p − 3) = 0 ⇒ p = 13, 3
3x = 13 or 3x = 3
x = −1, x = 1
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