9th grade algebra

Example

Expand and simplify: (x − 9)(x − 4)

1. x² − 4x − 9x + 36
2. x² − 13x + 36

Answer: x^2-13x+36

Example

Factorise: x² + 15x + 56

1. 8 × 7 = 56 and 8 + 7 = 15
2. (x + 8)(x + 7)

Answer: (x + 8)(x + 7)

Example

Factorise: 9x² − 16

1. (3x + 4)(3x − 4)

Answer: (3x + 4)(3x − 4)

Example

Solve: x² + 7x − 18 = 0. Give both solutions, separated by a comma.

1. (x − 2)(x + 9) = 0
2. x = 2 or x = −9

Answer: x = -9 or x = 2

Example

Make x the subject of y = 9x − 2. Type the expression for x.

1. y + 2 = 9x
2. x = (y + 2)/9

Answer: (y+2)/9

Example

f(x) = 2x² + 7. Find f(0).

1. f(0) = 2 × 0² + 0 × 0 + 7
2. = 7

Answer: 7

Example

y = 17 × 1.1ˣ. Find y when x = 4. Round to 2 decimal places if needed.

1. 1.1⁴ = 1.4641
2. 17 × 1.4641 = 24.8897

Answer: 24.89

Example

Adult tickets cost 10 and child tickets cost 4. A school sold 35 tickets and took 206 in total. How many adult tickets were sold?

1. c = 35 − a
2. 10a + 4(35 − a) = 206
3. 6a = 66
4. a = 11

Answer: 11

Example

Does the parabola y = −(x + 4)² − 6 have a maximum or a minimum point?

1. Negative coefficient: opens down, so a maximum.

Answer: Maximum

Example

Solve: x/4 + 4 = 10

1. x/4 = 6
2. x = 6 × 4
3. x = 24

Answer: x = 24

Example

Rearrange x + 4y = −24 to make y the subject. Type the expression for y.

1. 4y = −24 − x
2. y = −(1/4)x − 6

Answer: -(1/4)x-6

Example

How many solutions does 6(x + 5) + 8 = 6x + 38 have?

1. 6x + 38 = 6x + 38
2. Both sides are identical: infinitely many solutions.
3. Infinitely many solutions

Answer: Infinitely many solutions

Example

Solve: x + 8 ≥ 15

1. x ≥ 7
2. x ≥ 7

Answer: x ≥ 7

Example

Ethan solved an inequality: 1) −4x − 3 ≤ 1 2) −4x ≤ 4 3) x ≤ −1 Which line has the first mistake?

1. Dividing by −4 must reverse the sign: x ≥ −1.
2. Line 3

Answer: Line 3

Example

Solve: −13 < 4x − 1 < 7

1. −12 < 4x < 8
2. −3 < x < 2

Answer: −3 < x < 2

Example

Solve: x − 1 < −1 or 5x − 2 > 18

1. x < 0
2. 5x > 20 → x > 4
3. x < 0 or x > 4

Answer: x < 0 or x > 4

Example

Solve: |x − 7| = 9. Give both solutions, separated by a comma.

1. x − 7 = 9 → x = 16
2. x − 7 = −9 → x = −2
3. x = −2, x = 16

Answer: x = -2 or x = 16

Example

How many solutions does |x + 1| = 8 have?

1. Positive: two solutions.
2. 2

Answer: 2

Example

Is this relation a function? {(−1, 7), (7, −3), (−3, 5), (−3, 1), (4, 2)}

1. x = −3 has two different outputs: not a function.
2. No

Answer: No

Example

R = {(9, 0), (7, −5), (3, −4), (−1, 1)}. What is the domain of R?

1. {−1, 3, 7, 9}

Answer: {−1, 3, 7, 9}

Example

f(x) = 2x − 1 and g(x) = x² + 1. Find f(−2) − g(3).

1. f(−2) = −5
2. g(3) = 10
3. −5 − 10 = −15

Answer: -15

Example

x: 0, 1, 2, 3, 4 f(x): 2, 8, 20, 38, 62 Find the average rate of change of f from x = 0 to x = 4.

1. (62 − 2) ÷ (4 − 0)
2. = 15

Answer: 15

Example

What is the slope (gradient) of the line 3x + y = 0?

1. y = −3x
2. y = −3x
3. m = −3

Answer: -3

Example

Find the y-intercept of 2x − y = 6. Type it like (4, 0).

1. −y = 6 → y = −6
2. (0, −6)

Answer: (0, −6)

Example

Which equation is the line through (4, −1) with slope 1?

1. y + 1 = (x − 4)

Answer: y + 1 = (x − 4)

Example

Find the equation of the line through (−3, −5) and (3, −1). Type the right-hand side of y = … (fractions are fine).

1. m = 4/6 = 2/3
2. y = (2/3)x − 3

Answer: (2/3)x-3

Example

Find the line parallel to −3x − y = −1 through (−1, 6). Type the right-hand side of y = … .

1. m = −3
2. b = 6 − −3 × (−1) = 3
3. y = −3x + 3

Answer: -3x+3

Example

A line has slope 1/2. What is the slope of a line perpendicular to it? Give a fraction or whole number.

1. 1/2 × m = −1
2. m = −2

Answer: -2

Example

Line 1: y = (1/3)x + 5 Line 2: y = (1/3)x + 6 Are the lines parallel, perpendicular or neither?

1. 1/3 and 1/3
2. Parallel

Answer: Parallel

Example

Solve the system: y = −3x − 16 4x − 3y = −17 Type the solution like (2, −1).

1. 4x − 3(−3x − 16) = −17
2. 13x = −65 → x = −5
3. y = −1
4. (−5, −1)

Answer: (−5, −1)

Example

Solve: −5x + y = −15 4x − y = 13 Type the solution like (2, −1).

1. −x = −2 → x = 2
2. y = −5
3. (2, −5)

Answer: (2, −5)

Example

Solve: x − 3y = 15 −5x − 6y = 30 Type the solution like (2, −1).

1. 2x − 6y = 30
2. 7x = 0 → x = 0
3. y = −5
4. (0, −5)

Answer: (0, −5)

Example

How many solutions does this system have? y = x + 4 y = x + 7

1. y = x + 7
2. None

Answer: None

Example

The sum of two numbers is 38 and their difference is 10. What is the larger number?

1. 2x = 48
2. x = 24

Answer: 24

Example

Write 2⁻² as a fraction.

1. 2⁻² = 1/2²
2. = 1/4

Answer: 1/4

Example

Simplify: (2x⁴)(2x)

1. 2 × 2 = 4
2. x⁴ × x = x⁵
3. 4x⁵

Answer: 4x^5

Example

Simplify: −16x³ ÷ 4x²

1. −16 ÷ 4 = −4
2. x³ ÷ x² = x
3. −4x

Answer: -4x

Example

Simplify: (−3x⁵)²

1. (−3)² = 9
2. (x⁵)² = x¹⁰
3. 9x¹⁰

Answer: 9x^10

Example

What is the degree of the polynomial −3x² + 6x + 8?

1. Highest power: x²
2. 2

Answer: 2

Example

Simplify: (−6x² + 5x + 9) + (2x² − 3x − 6)

1. −4x² + 2x + 3

Answer: -4x^2+2x+3

Example

Expand: 6x(4x² − 6x + 4)

1. 24x³ − 36x² + 24x

Answer: 24x^3-36x^2+24x

Example

Complete the factorisation: 6x⁴ + 10x³ − 4x² = 2x²( … ). Type what goes inside the bracket.

1. 6x⁴ + 10x³ − 4x² = 2x²(3x² + 5x − 2)
2. 3x² + 5x − 2

Answer: 3x^2+5x-2

Example

x² − 2x − 15 = (x + 3)(x + ☐). What number goes in the box?

1. 3 × (−5) = −15, 3 + (−5) = −2
2. −5

Answer: -5

Example

Factorise by grouping: x³ + 3x² − 2x − 6

1. x²(x + 3) − 2(x + 3)
2. (x² − 2)(x + 3)

Answer: (x² − 2)(x + 3)

Example

Factorise: x² − 4x + 4

1. 4 = 2², 2 × (−2) = −4
2. (x − 2)²

Answer: (x − 2)²

Example

Factorise fully: 2x² + 8x + 6

1. 2(x² + 4x + 3)
2. 2(x + 1)(x + 3)

Answer: 2(x + 1)(x + 3)

Example

Solve: (x − 8)(x − 7) = 0. Give both solutions, separated by a comma.

1. x − 8 = 0 or x − 7 = 0
2. x = 8 or x = 7

Answer: x = 7 or x = 8

Example

Solve: 2x² − 17 = 15. Give both solutions, separated by a comma.

1. x² = 16
2. x = ±4

Answer: x = -4 or x = 4

Example

Find the axis of symmetry of y = −2x² + 16x − 23. Give the value of x.

1. x = −(16) ÷ (2 × (−2))
2. x = 4

Answer: 4

Example

Which method is the most efficient way to solve (x + 6)² = 10?

1. Take square roots

Answer: Take square roots

Example

y = 350(0.92)ᵗ. Which describes the change each time period?

1. 0.92 = 1 − 0.08
2. decay of 8%

Answer: decay of 8%

Example

x: 0, 1, 2, 3, 4 y: 1, 3, 9, 27, 81 Is the relationship linear, exponential or neither?

1. Constant ratio: exponential.
2. Exponential

Answer: Exponential

Example

The half-life of caffeine in the body is 3 hours. Starting with 44 mg, how much is left after 6 hours?

1. 6 ÷ 3 = 2
2. 44 × (1/2)² = 11 mg

Answer: 11

Example

An arithmetic sequence starts 11, 17, 23, … Find the 75th term.

1. d = 6
2. a₇₅ = 11 + 74 × 6 = 455

Answer: 455

Example

What is the common ratio of the geometric sequence 1, 4, 16, 64, …?

1. 4 ÷ 1 = 4

Answer: 4

Example

Is 2, 6, 18, 54, 162, … arithmetic, geometric or neither?

1. Geometric

Answer: Geometric