Quadratic Equations

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수업 노트

Solving Quadratics by Factorising

Rearrange into ax² + bx + c = 0 with zero on one side.
Factorise the quadratic and set each bracket equal to zero.
If (x + 4)(x - 1) = 0, then x + 4 = 0 or x - 1 = 0.
For brackets with coefficients, e.g. (2x - 3)(3x + 5) = 0, solve 2x - 3 = 0 and 3x + 5 = 0.
If x is a factor, e.g. x(x - 4) = 0, solutions are x = 0 and x = 4.
Do not divide both sides by x; you will lose a solution.
Use a calculator to check factorisation: if solutions are integers or fractions, the quadratic factorises.

The Quadratic Formula

Formula: x = (-b ± √(b² - 4ac)) / (2a) for ax² + bx + c = 0 (a ≠ 0).
Read off a, b, c and substitute carefully, using brackets for negative numbers.
Simplify using a calculator or by hand; round as required (e.g. 2 d.p., 3 s.f.).
The discriminant is b² - 4ac: if > 0 two solutions, = 0 one solution, < 0 no real solutions.
If the discriminant is a perfect square, the quadratic factorises with integers.
Always show working; calculators can be used to check answers.

Completing the Square

Rewrite x² + bx as (x + p)² - p² where p = b/2.
For x² + bx + c, complete square: (x + p)² - p² + c, then simplify numbers.
If coefficient a ≠ 1, factorise a out of x² and x terms first: a[x² + (b/a)x] + c.
Then complete square inside brackets and multiply through by a.
The turning point of y = (x + p)² + q is at (-p, q); for y = a(x + p)² + q, same coordinates.
Turning point is minimum if a > 0, maximum if a < 0.
Check your answer by expanding the completed square.

Solving by Completing the Square

To solve x² + bx + c = 0, complete square: (x + p)² - p² + c = 0.
Rearrange to (x + p)² = p² - c, then take square roots: x + p = ±√(p² - c).
Solve for x: x = -p ± √(p² - c).
If a ≠ 1, divide both sides by a first (only for solving, not rewriting).
Answers are often in exact (surd) form.
Do not expand the squared bracket back out when solving.

Deciding the Quadratic Method

Use factorisation when the question says 'solve by factorising' or for simple quadratics.
Use quadratic formula when answers need a given accuracy (e.g. 2 d.p.) or when factorisation is hard.
Use completing the square when part (a) asks to complete the square and part (b) uses it to solve.
Completing the square also helps rearrange formulae with x² and x terms.
If in doubt, the quadratic formula always works.
Check solutions with a calculator: integer/fraction solutions mean factorisation works.

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연습 문제

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1.What is the standard form of a quadratic equation?
Easy
- Aax + bx + c = 0
- Bax² + bx + c = 0
- Cax² + bx + c = y
- Dax + b = 0
2.Solve x² - 4x = 0 by factorising.
Easy
- Ax = 0 or x = 4
- Bx = 0 or x = -4
- Cx = 2 or x = -2
- Dx = 4 only
3.Which of the following is the quadratic formula?
Easy
- Ax = (-b ± √(b² - 4ac)) / (2a)
- Bx = (-b ± √(b² + 4ac)) / (2a)
- Cx = (b ± √(b² - 4ac)) / (2a)
- Dx = (-b ± √(b² - 4ac)) / a
4.Use the quadratic formula to solve 3x² + 7x - 11 = 0. Give your answers correct to 2 decimal places.
Medium
- Ax = 1.08 or x = -3.41
- Bx = 1.08 or x = 3.41
- Cx = -1.08 or x = 3.41
- Dx = -1.08 or x = -3.41
5.By completing the square, solve x² + 6x + 5 = 0.
Medium
- Ax = -1 or x = -5
- Bx = 1 or x = 5
- Cx = -1 or x = 5
- Dx = 1 or x = -5
6.Write x² - 4x + 7 in the form (x - a)² + b.
Medium
- A(x - 2)² + 3
- B(x - 2)² + 7
- C(x - 4)² + 7
- D(x - 2)² + 11
7.The discriminant of 2x² - 3x + 1 = 0 is:
Easy
- A1
- B9
- C-1
- D17
8.Solve 10m² + 9m - 162 = 0.
Medium
- Am = 3.6 or m = -4.5
- Bm = -3.6 or m = 4.5
- Cm = 3.6 or m = 4.5
- Dm = -3.6 or m = -4.5

더 많은 문제

9.The nth term of a sequence is 4n² + n + 3. Find n when the nth term is 498.
Hard
- An = 11
- Bn = 12
- Cn = 10
- Dn = 13
10.Write x² + 8x - 9 in the form (x + k)² + h.
Medium
- A(x + 4)² - 25
- B(x + 4)² - 9
- C(x + 8)² - 9
- D(x + 4)² + 7
11.Solve 2x² + 9x - 7 = 0. Give your solutions correct to 3 significant figures.
Medium
- Ax = 0.676 or x = -5.18
- Bx = 0.676 or x = 5.18
- Cx = -0.676 or x = 5.18
- Dx = -0.676 or x = -5.18
12.How many real solutions does x² + 2x + 5 = 0 have?
Medium
- A0
- B1
- C2
- DCannot be determined
13.Solve by factorisation: 10r² - 23r + 9 = 0.
Medium
- Ar = 0.5 or r = 1.8
- Br = 0.5 or r = -1.8
- Cr = -0.5 or r = 1.8
- Dr = -0.5 or r = -1.8
14.The solutions of x² + bx + c = 0 are (-7 + √61)/2 and (-7 - √61)/2. Find b and c.
Hard
- Ab = 7, c = -3
- Bb = 7, c = 3
- Cb = -7, c = -3
- Db = -7, c = 3
15.What is the turning point of y = x² + 6x + 4?
Medium
- A(-3, -5)
- B(3, -5)
- C(-3, 4)
- D(3, 4)
16.Solve 2x² - 8x - 24 = 0 by completing the square.
Medium
- Ax = 6 or x = -2
- Bx = 6 or x = 2
- Cx = -6 or x = 2
- Dx = -6 or x = -2
17.Solve x² - 5x + 6 = 0 by factorisation.
Easy
- Ax = 2 or x = 3
- Bx = -2 or x = -3
- Cx = 1 or x = 6
- Dx = -1 or x = -6
18.Solve 2x² + 5x - 3 = 0 by factorisation.
Medium
- Ax = 0.5 or x = -3
- Bx = -0.5 or x = 3
- Cx = 1 or x = -1.5
- Dx = -1 or x = 1.5
19.Use the quadratic formula to solve 9x² - 12x - 23 = 0. Give your answers correct to 2 decimal places.
Medium
- Ax = 2.40 or x = -1.07
- Bx = 2.40 or x = 1.07
- Cx = -2.40 or x = 1.07
- Dx = -2.40 or x = -1.07
20.Solve 3x² - 2x - 10 = 0. Give your answers correct to 2 decimal places.
Medium
- Ax = 2.19 or x = -1.52
- Bx = 2.19 or x = 1.52
- Cx = -2.19 or x = 1.52
- Dx = -2.19 or x = -1.52
21.Solve 3x² - 2x - 2 = 0. Give your answers correct to 2 decimal places.
Medium
- Ax = 1.22 or x = -0.55
- Bx = -1.22 or x = 0.55
- Cx = 1.22 or x = 0.55
- Dx = -1.22 or x = -0.55
22.Solve 6m² + 25m + 16 = 0. Give your answers correct to 2 decimal places.
Medium
- Am = -0.79 or m = -3.38
- Bm = 0.79 or m = 3.38
- Cm = -0.79 or m = 3.38
- Dm = 0.79 or m = -3.38
23.Solve 2x² + 7x - 3 = 0. Give your answers correct to 2 decimal places.
Medium
- Ax = 0.39 or x = -3.89
- Bx = -0.39 or x = 3.89
- Cx = 0.39 or x = 3.89
- Dx = -0.39 or x = -3.89
24.Write x² - 18x - 27 in the form (x + k)² + h.
Medium
- A(x - 9)² - 108
- B(x - 9)² + 54
- C(x + 9)² - 108
- D(x - 18)² - 351
25.Using your answer to part (a) (or otherwise), solve x² - 18x - 27 = 0. Give your answers in exact form.
Medium
- Ax = 9 \pm 6\√{3}
- Bx = 9 \pm 3\√{6}
- Cx = 9 \pm 6\√{6}
- Dx = 9 \pm 3\√{3}
26.The area of a trapezium is 351 cm². The parallel sides are (x+7) cm and (x+2) cm, and the perpendicular height is 2x cm. Which equation represents this situation?
Medium
- A2x² + 9x - 351 = 0
- B2x² + 9x + 351 = 0
- C2x² + 9x - 702 = 0
- D2x² + 9x + 702 = 0
27.The area of a square is 30 cm². The side length is (x+5) cm. Which equation represents this?
Medium
- Ax² + 10x + 25 = 30
- Bx² + 10x + 25 = 0
- Cx² + 10x - 5 = 0
- Dx² + 10x + 5 = 0
28.The difference between the areas of two rectangles is 62 cm². The first rectangle has length (x+5) cm and width (x-2) cm. The second rectangle has length (x+2) cm and width (x-5) cm. Which equation represents this situation?
Medium
- A(x+5)(x-2) - (x+2)(x-5) = 62
- B(x+2)(x-5) - (x+5)(x-2) = 62
- C(x+5)(x-2) + (x+2)(x-5) = 62
- D(x+5)(x-2) = 62
29.What is the value of the discriminant for the quadratic equation x² + 4x + 4 = 0?
Easy
- A0
- B16
- C8
- D-8
30.Which of the following is the correct way to complete the square for x² + 6x?
Easy
- A(x+3)² - 9
- B(x+3)² + 9
- C(x+6)² - 36
- D(x+3)² - 3
31.The quadratic equation 2x² + 3x + 5 = 0 has two real solutions.
Easy
True or false?
32.Which of the following are methods that can be used to solve a quadratic equation? (Select all that apply)
Easy
- AFactorisation
- BQuadratic formula
- CCompleting the square
- DLinear interpolation
- ETrial and improvement
33.Match each quadratic equation to its most appropriate solving method.
Medium
- x² + 5x + 6 = 0
- x² + 2x - 5 = 0 (answers to 2 d.p.)
- x² + 4x + 1 = 0 (exact form)
- Factorisation
- Quadratic formula
- Completing the square
34.Put the following steps in the correct order to solve x² + 6x + 8 = 0 by completing the square.
Medium
- (x+3)² - 9 + 8 = 0
- (x+3)² = 1
- x+3 = \pm 1
- x = -2 or x = -4
- Write x²+6x as (x+3)² - 9
35.The equation x² + kx + 9 = 0 has exactly one real solution. What is the value of k?
Medium
- A6
- B-6
- C6 or -6
- D3
36.By completing the square, find the turning point of y = x² - 4x + 7.
Medium
- A(2, 3)
- B(-2, 3)
- C(2, 7)
- D(-2, 7)
37.Solve x² - 2x - 15 = 0 by factorisation. What are the solutions?
Easy
- Ax = 5 or x = -3
- Bx = -5 or x = 3
- Cx = 5 or x = 3
- Dx = -5 or x = -3
38.Use the quadratic formula to solve 2x² + 3x - 4 = 0. Which of the following gives the correct solutions, correct to 2 decimal places?
Medium
- Ax = 0.85 or x = -2.35
- Bx = 0.85 or x = 2.35
- Cx = -0.85 or x = -2.35
- Dx = -0.85 or x = 2.35
39.Solve x² - 9 = 0 by factorisation. Enter the positive solution.
Easy
40.Complete the sentence about the quadratic formula.
Easy
For a quadratic equation ax² + bx + c = 0, the solutions are given by x = \frac{-b \pm \√{____}}{2a}.
41.Write down the value of the discriminant of x² + 2x + 5 = 0 and state how many real roots the equation has.
Medium
42.Solve 2x² - 7x + 3 = 0 by factorisation. Enter the smaller solution.
Medium
43.Solve x² - 25 = 0 by factorisation. Enter the positive solution.
Easy
44.Solve x² - 9 = 0 by factorisation. Enter the positive solution.
Easy
- A2
- B5
- C4
- D3
45.Solve 2x² - 7x + 3 = 0 by factorisation. Enter the smaller solution.
Medium
- A1.5
- B2.5
- C-0.5
- D0.5
46.Solve x² - 25 = 0 by factorisation. Enter the positive solution.
Easy
- A6
- B7
- C5
- D4

복습 노트

## Quadratic Equations - Revision Notes ### 1. Solving by Factorising - Rearrange to **ax² + bx + c = 0**. - Factorise: if (x + p)(x + q) = 0, then x = -p or x = -q. - For coefficients: (2x - 3)(3x + 5) = 0 ⇒ 2x - 3 = 0 or 3x + 5 = 0 ⇒ x = 3/2 or x = -5/3. - If x is a factor: x(x - 4) = 0 ⇒ x = 0 or x = 4. **Do not divide by x**. ### 2. Quadratic Formula - For ax² + bx + c = 0 (a ≠ 0): \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] - **Discriminant** Δ = b² - 4ac: - Δ > 0: two distinct real solutions. - Δ = 0: one repeated solution. - Δ < 0: no real solutions. - Use when answers required to a given accuracy (e.g., 2 d.p.). ### 3. Completing the Square - Rewrite x² + bx as (x + p)² - p², where p = b/2. - For ax² + bx + c: factor a out of first two terms, complete square, simplify. - **Turning point**: y = a(x + p)² + q has vertex at (-p, q). - To solve: (x + p)² = q ⇒ x = -p ± √q. ### 4. Choosing a Method - **Factorising**: quick if factorable; use for simple equations. - **Quadratic formula**: always works; use for decimals/surds. - **Completing the square**: when directed or for deriving turning point. ### Key Points - Always ensure equation = 0 before solving. - Check calculator solutions to verify factorisation. - For exact surd form, simplify √(b² - 4ac) fully.

Quadratic equation: An equation of the form ax² + bx + c = 0, where a ≠ 0.
Factorisation: Writing a quadratic as a product of two linear factors, e.g., (x + p)(x + q) = 0.
Quadratic formula: x = (-b ± √(b² - 4ac)) / (2a), used to solve any quadratic equation.
Discriminant: The value b² - 4ac under the square root in the quadratic formula; determines number of real solutions.
Completing the square: Rewriting ax² + bx + c as a(x + p)² + q to find turning point or solve.
Turning point: The vertex of a quadratic graph; minimum if a > 0, maximum if a < 0.

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