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The quadratic formula practice questions with answers

Higher · Grade 7 · Calculator

A second set of the quadratic formula questions for GCSE Maths. They are different from the ones on the topic page, so work through both. Try each question, then open the answer to check your working.

More questions

Question 1 Easy

x² + 6x - 2 = 0 will not factor nicely. Write the exact roots from the quadratic formula.

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Answer: -3 ± √11

  1. Discriminant 36 + 8 = 44 = 4 × 11.
  2. (-6 ± 2√11) / 2 = -3 ± √11.
  • Method (1): Discriminant 36 + 8 = 44 = 4 × 11.
  • Accuracy (1): (-6 ± 2√11) / 2 = -3 ± √11.

Question 2 Easy

Using the quadratic formula, what is the discriminant of x² + 3x + 1 = 0?

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Answer: 5

  1. Square 3 to get 9.
  2. 4ac is 4, so the discriminant is 5.
  • Method (1): Square 3 to get 9.
  • Accuracy (1): 4ac is 4, so the discriminant is 5.

Question 3 Medium

Find the exact roots of 2x² + x - 2 = 0 using the quadratic formula.

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Answer: (-1 ± √17)/4

  1. Discriminant 1 + 16 = 17.
  2. (-1 ± √17) / 4.
  • Method (1): Discriminant 1 + 16 = 17.
  • Accuracy (1): (-1 ± √17) / 4.

Question 4 Medium

How many real roots does x² + 2x + 8 = 0 have? Use the discriminant from the quadratic formula.

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Answer: 0

  1. Discriminant 4 - 32 = -28.
  2. A negative discriminant means no real root.
  • Method (1): Discriminant 4 - 32 = -28.
  • Accuracy (1): A negative discriminant means no real root.

Question 5 Hard

Work out the exact roots of 2x² + 4x - 1 = 0 and cancel the common factor of 2.

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Answer: (-2 ± √6)/2

  1. Discriminant 16 + 8 = 24 = 4 × 6.
  2. (-4 ± 2√6) / 4 = (-2 ± √6) / 2.
  • Method (1): Discriminant 16 + 8 = 24 = 4 × 6.
  • Accuracy (1): (-4 ± 2√6) / 4 = (-2 ± √6) / 2.

Question 6 Hard

The quadratic formula on x² - x - 6 = 0 gives integer roots. State the positive root only.

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Answer: 3

  1. The equation factors as (x - 3)(x + 2) = 0.
  2. The positive root is 3.
  • Method (1): The equation factors as (x - 3)(x + 2) = 0.
  • Accuracy (1): The positive root is 3.

Exam-style question

A longer question with marks, in the style of a written paper. This is a Yumio practice question, not a past paper question.

Exam-style question 13 marks

Use the quadratic formula to solve x² + 8x - 4 = 0, giving exact simplified roots.

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Answer: -4 ± 2√5

  1. a = 1, b = 8, c = -4.
  2. Discriminant 64 + 16 = 80 = 16 × 5.
  3. (-8 ± 4√5) / 2.
  4. -4 ± 2√5.
  • Method (1): Substituted a = 1, b = 8, c = -4 into the formula
  • Method (1): Discriminant 80, or √80 = 4√5
  • Accuracy (1): -4 ± 2√5

Quick checks

Short one-step questions. Pick an option before you open the answer.

Check 1

What is the quadratic formula for ax² + bx + c = 0?

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x = (-b ± √(b² - 4ac)) / (2a). x = (-b ± √(b² - 4ac)) / (2a).

Check 2

Why keep √17 in the answer for 2x² + 3x - 1 = 0?

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17 is not a square, so the exact roots are (-3 ± √17) / 4. 17 is not a square, so the exact roots are (-3 ± √17) / 4.

Check 3

What does a negative discriminant tell you?

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The equation has no real roots. The curve misses the x-axis. The equation has no real roots. The curve misses the x-axis.

Check 4

What is 2a when the equation starts with 2x²?

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4. The denominator of the formula is 4, not 2. 4. The denominator of the formula is 4, not 2.

Check 5

How do you simplify (2 ± 2√2) / 2?

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Divide every term by 2. That simplifies to 1 ± √2. Divide every term by 2. That simplifies to 1 ± √2.

Check 6

Why is the root of x² - 3 = 0 equal to ±√3, not ±√12?

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The formula produces ±√12 / 2, and √12 / 2 simplifies to √3. The formula produces ±√12 / 2, and √12 / 2 simplifies to √3.

Check 7

Which sign does b have in x² - 2x - 1 = 0?

  1. b is -2. The minus belongs to the coefficient.
  2. 17 is not a square, so the exact roots are (-3 ± √17) / 4.
  3. 3. The other root is -2.
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b is -2. The minus belongs to the coefficient. b is -2. The minus belongs to the coefficient.

Check 8

What is b² - 4ac for x² + 3x + 1 = 0?

  1. 9 - 4 = 5.
  2. b is -2. The minus belongs to the coefficient.
  3. 3. The other root is -2.
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9 - 4 = 5. 9 - 4 = 5.

Check 9

Why write two answers when you see ±?

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The square root has a positive and a negative value, unless it is zero. The square root has a positive and a negative value, unless it is zero.

Check 10

A student writes 0.25 times root 17. What exact fraction is that coefficient?

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√17 / 4. Do not replace the fraction with a decimal. √17 / 4. Do not replace the fraction with a decimal.

Check 11

Solve x² - 5x + 6 = 0 with the formula. What are the roots?

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2 or 3. The discriminant 25 is a square, so the surd disappears. 2 or 3. The discriminant 25 is a square, so the surd disappears.

Check 12

What is the first step before substituting into the formula?

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Write down a, b and c, including their signs. Write down a, b and c, including their signs.

Check 13

Why is (1 ± √7) / 3 better than (2 ± 2√7) / 6?

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A common factor of 2 has been cancelled. The value is unchanged. A common factor of 2 has been cancelled. The value is unchanged.

Check 14

The equation x² + 2x + 8 = 0 has discriminant -28. How many real roots?

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None. You cannot take the square root of a negative number here. None. You cannot take the square root of a negative number here.

Check 15

What is the positive root of x² - x - 6 = 0?

  1. 9 - 4 = 5.
  2. 3. The other root is -2.
  3. b is -2. The minus belongs to the coefficient.
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3. The other root is -2. 3. The other root is -2.

Keep going with The quadratic formula

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