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课程笔记
Solving by factorising
- A quadratic equation has form ax² + bx + c = 0 with a ≠ 0. Rearrange one side to zero before using the zero-product method.
- Factorise and use the fact that if AB = 0 then A = 0 or B = 0. x2 - 5x + 6 = 0 gives (x - 2)(x - 3) = 0, so x = 2 or x = 3.
- For ax2 + bx + c with a \neq 1, find two numbers that multiply to ac and add to b, then split the middle term.
- x2 - 49 = 0 is a difference of two squares: (x - 7)(x + 7) = 0.
The quadratic formula
- When a quadratic will not factorise easily, use x = \frac{-b \pm \√{b2 - 4ac}}{2a}.
- Substitute a, b and c carefully, using brackets for negative values.
- The part under the root, b2 - 4ac, is the discriminant: positive gives two roots, zero gives one repeated root, negative gives no real roots.
Completing the square
- Write x2 + bx + c as (x + \frac{b}{2})2 + c - (\frac{b}{2})2.
- x² + 6x + 2 = (x + 3)² − 7. For the equation x² + 6x + 2 = 0, this gives (x + 3)² = 7 and x = −3 ± √7.
- The turning point of (x + p)2 + q is (-p, q).
Common errors
- Not setting the equation equal to zero first (x2 + 3x = 10 is not x(x + 3) = 10 \Rightarrow x = 10).
- Losing the second solution when taking a square root: x2 = 9 gives x = \pm 3.
- Sign slips when b is negative in the formula: -b becomes positive.
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练习题
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1.What must a quadratic equation equal before you factorise it to solve?
Easy2.Solve x2 - 5x + 6 = 0.
Easy- Ax = 2, 3
- Bx = -2, -3
- Cx = 1, 6
- Dx = -1, -6
- Ax = 5
- Bx = \pm 5
- Cx = 12.5
- Dx = \pm 12.5
4.Solve x2 + 7x + 10 = 0.
Easy- Ax = 2, 5
- Bx = -2, -5
- Cx = 1, 10
- Dx = -1, -10
5.Solve x2 - x - 6 = 0.
Easy- Ax = 3, -2
- Bx = -3, 2
- Cx = 6, -1
- Dx = 1, -6
- Ax = 7
- Bx = \pm 7
- Cx = \pm 49
- Dx = 24.5
- Ax = 0, -4
- Bx = 4
- Cx = 0, 4
- Dx = -4
8.Solve x2 + 3x = 10.
Medium- Ax = 2, -5
- Bx = -2, 5
- Cx = 3, 7
- Dx = 10, -1
更多题目
9.Solve 2x2 + 5x + 2 = 0.
Medium- Ax = -2, -\frac{1}{2}
- Bx = 2, \frac{1}{2}
- Cx = -1, -4
- Dx = 1, 4
10.Solve 3x2 - 7x + 2 = 0.
Medium- Ax = 2, \frac{1}{3}
- Bx = -2, -\frac{1}{3}
- Cx = 3, \frac{2}{3}
- Dx = 1, 6
11.What is the discriminant of x2 - 6x + 9 = 0?
Medium12.What is the discriminant of x2 + 3x + 4 = 0?
Medium13.What does a discriminant of -7 tell you?
Medium- Atwo real roots
- Bone real root
- Cno real roots
- Dthree roots
14.Complete the square: x2 + 6x + 2.
Medium- A(x + 3)2 - 7
- B(x + 3)2 + 2
- C(x + 6)2 - 34
- D(x + 3)2 - 9
15.Complete the square: x2 - 8x + 3.
Medium- A(x - 4)2 - 13
- B(x - 4)2 + 3
- C(x - 8)2 - 61
- D(x - 4)2 + 13
16.Solve x2 + 4x - 3 = 0 giving answers to 2 decimal places.
Medium- Ax = 0.65, -4.65
- Bx = 1, -3
- Cx = 0.46, -4.46
- Dx = 3.65, -0.65
17.Use the formula to solve 2x2 - 3x - 4 = 0 (2 d.p.).
Medium- Ax = 2.35, -0.85
- Bx = 1.5, -4
- Cx = 2.5, -0.5
- Dx = 3, -1
18.The roots of x2 + bx + 12 = 0 are 3 and 4. What is b?
Medium19.The area of a rectangle is 24 cm². The length is 2 cm more than the width x. Which equation gives x?
Medium- Ax2 + 2x - 24 = 0
- Bx2 - 2x - 24 = 0
- C2x2 = 24
- Dx + 2 = 24
20.For the rectangle x(x + 2) = 24, what is x?
Medium21.x2 - 6x + 11 written as (x - p)2 + q has q equal to:
Medium22.Why must you consider both + and - when solving x2 = 36?
Easy- Abecause (-6)2 = 36 too
- Bbecause 36 is even
- Cto make the answer integer
- Dthere is no reason
23.When is the quadratic formula most useful?
Medium- Awhen the quadratic factorises easily
- Bwhen it will not factorise easily
- Cwhen there is no x2
- Dfor linear equations
24.Which is the correct quadratic formula?
Easy- A\frac{b \pm \√{b2 - 4ac}}{2a}
- B\frac{-b \pm \√{b2 - 4ac}}{2a}
- C\frac{-b \pm \√{b2 + 4ac}}{2a}
- D\frac{-b \pm \√{b2 - 4ac}}{a}
25.x2 - 4x + k = 0 has one repeated root. What is k?
Medium26.Solving x2 = 49 gives x = 7 only.
EasyTrue or false?
27.A quadratic equation can have two different solutions.
EasyTrue or false?
28.If the discriminant is negative, the quadratic has no real roots.
EasyTrue or false?
29.x2 + 3x = 10 can be solved by writing x(x + 3) = 10 so x = 10.
EasyTrue or false?
30.The turning point of (x + 2)2 - 5 is (2, -5).
MediumTrue or false?
31.Which equations have solutions x = 2 and x = 3?
Medium- Ax2 - 5x + 6 = 0
- B(x - 2)(x - 3) = 0
- Cx2 + 5x + 6 = 0
- Dx2 - x - 6 = 0
- E2x2 - 10x + 12 = 0
32.Which have no real solutions?
Medium- Ax2 + 1 = 0
- Bx2 - 1 = 0
- Cx2 + 2x + 3 = 0
- Dx2 + 2x - 3 = 0
- Ex2 = 0
33.Solve x2 + 6x + 4 = 0 by completing the square, giving your answers in surd form.
Medium
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