Quadratics and factoring test

Factor it, solve it, spot the pattern, and know how many answers to expect before you start.

Quadratics and factoring test

The answer key prints on a separate page.

What's on this test

Factoring a quadratic is a search for two numbers that multiply to the constant and add to the middle coefficient, and the reliable error is finding a pair that does one but not the other. Solving follows directly once that is done — the zero-product rule says a product is zero only when a factor is, so the factors hand you the solutions. A child who can factor but not solve has stopped one step early, not misunderstood.

How the paper is put together

The difference of squares is the pattern worth recognizing on sight: a² − b² is always (a + b)(a − b), and a paper that mixes it in checks whether it is seen or ground out the long way. The discriminant section is deliberately last and asks a different kind of question — not what the solutions are but how many there are. Knowing a quadratic has no real solutions before spending five minutes hunting for one is the judgment the unit is really building.

Math test

Name: Date:

1. Factoring x² + bx + cFactor each one. Write the two numbers that go in the brackets.6 questions

1x² + 20x + 19(x + )(x + )

2x² + 11x + 24(x + )(x + )

3x² + 7x + 10(x + )(x + )

4x² + 34x + 285(x + )(x + )

5x² + 38x + 325(x + )(x + )

6x² + 32x + 175(x + )(x + )

2. Solving x² + bx + c = 0Each product is zero. Write the two values of x.5 questions

7(x + 18)(x + 7) = 0x = x =

8(x − 6)(x + 12) = 0x = x =

9(x − 11)(x − 15) = 0x = x =

10(x + 13)(x + 1) = 0x = x =

11(x − 16)(x + 17) = 0x = x =

3. How many real solutions?How many real solutions does each equation have? Circle one.5 questions

12x² − 8x + 4 = 0TwoOneNone

13x² − 13x + 20 = 0TwoOneNone

14x² + 12x + 36 = 0TwoOneNone

Math test

3. How many real solutions?How many real solutions does each equation have? Circle one.continued

15x² − 22x − 73 = 0TwoOneNone

16x² + 10x + 25 = 0TwoOneNone

4. Difference of two squaresEach one is a difference of two squares. Fill in the missing number.5 questions

17x² − 144(x + )(x − )

18x² − 1296(x + )(x − )

19x² − 441(x + )(x − )

20x² − 576(x + )(x − )

21x² − 529(x + )(x − )

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Answer key

Factoring x² + bx + c

1. x² + 20x + 19 = (x + 19)(x + 1) (19 and 1)2. x² + 11x + 24 = (x + 3)(x + 8) (3 and 8)3. x² + 7x + 10 = (x + 2)(x + 5) (2 and 5)4. x² + 34x + 285 = (x + 15)(x + 19) (15 and 19)5. x² + 38x + 325 = (x + 13)(x + 25) (13 and 25)6. x² + 32x + 175 = (x + 25)(x + 7) (25 and 7)

Solving x² + bx + c = 0

7. (x + 18)(x + 7) = 0 → (x + 18)(x + 7) = 0 → x = −18 or x = −78. (x − 6)(x + 12) = 0 → (x − 6)(x + 12) = 0 → x = 6 or x = −129. (x − 11)(x − 15) = 0 → (x − 11)(x − 15) = 0 → x = 11 or x = 1510. (x + 13)(x + 1) = 0 → (x + 13)(x + 1) = 0 → x = −13 or x = −111. (x − 16)(x + 17) = 0 → (x − 16)(x + 17) = 0 → x = 16 or x = −17

How many real solutions?

12. x² − 8x + 4 = 0: −8² − 4 × 1 × 4 = 64 − 16 = 48 → Two13. x² − 13x + 20 = 0: −13² − 4 × 1 × 20 = 169 − 80 = 89 → Two14. x² + 12x + 36 = 0: 12² − 4 × 1 × 36 = 144 − 144 = 0 → One15. x² − 22x − 73 = 0: −22² − 4 × 1 × −73 = 484 − -292 = 776 → Two16. x² + 10x + 25 = 0: 10² − 4 × 1 × 25 = 100 − 100 = 0 → One

Difference of two squares

17. x² − 144 = (x + 12)(x − 12) (144 is 12 squared, so the two brackets are (x + 12) and (x − 12))18. x² − 1296 = (x + 36)(x − 36) (1296 is 36 squared, so the two brackets are (x + 36) and (x − 36))19. x² − 441 = (x + 21)(x − 21) (441 is 21 squared, so the two brackets are (x + 21) and (x − 21))20. x² − 576 = (x + 24)(x − 24) (576 is 24 squared, so the two brackets are (x + 24) and (x − 24))21. x² − 529 = (x + 23)(x − 23) (529 is 23 squared, so the two brackets are (x + 23) and (x − 23))

Quadratics and factoring test

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