Grade 6 Math with Ms. Eringis : Answer Key--Quadratic Formula Practice - Free Printable

Let’s solve each problem one by one using the quadratic formula:

The quadratic formula is:
> x = [ -b ± √(b² - 4ac) ] / (2a)

We’ll apply this to each equation. Remember: first write the equation in standard form: ax² + bx + c = 0.

---

Problem 1: x² + 10x - 2 = 0

Here, a = 1, b = 10, c = -2

Discriminant = b² - 4ac = 10² - 4(1)(-2) = 100 + 8 = 108

√108 ≈ 10.392

So,

x = [-10 ± 10.392] / 2

First solution: (-10 + 10.392)/2 = 0.392/2 = 0.196 ≈ 0.20

Second solution: (-10 - 10.392)/2 = -20.392/2 = -10.196 ≈ -10.20

✔ Final Answer for #1: {0.20, -10.20}

---

Problem 2: x² - 11 = 4x → rewrite as x² - 4x - 11 = 0

a = 1, b = -4, c = -11

Discriminant = (-4)² - 4(1)(-11) = 16 + 44 = 60

√60 ≈ 7.746

x = [4 ± 7.746]/2

First: (4 + 7.746)/2 = 11.746/2 = 5.873 ≈ 5.87

Second: (4 - 7.746)/2 = -3.746/2 = -1.873 ≈ -1.87

✔ Final Answer for #2: {5.87, -1.87}

---

Problem 3: x² - 8x = 20 → rewrite as x² - 8x - 20 = 0

a = 1, b = -8, c = -20

Discriminant = (-8)² - 4(1)(-20) = 64 + 80 = 144

√144 = 12

x = [8 ± 12]/2

First: (8 + 12)/2 = 20/2 = 10

Second: (8 - 12)/2 = -4/2 = -2

✔ Final Answer for #3: {10, -2}

---

Problem 4: x² - 5x - 36 = 0

a = 1, b = -5, c = -36

Discriminant = (-5)² - 4(1)(-36) = 25 + 144 = 169

√169 = 13

x = [5 ± 13]/2

First: (5 + 13)/2 = 18/2 = 9

Second: (5 - 13)/2 = -8/2 = -4

✔ Final Answer for #4: {9, -4}

---

Problem 5: x² + 6x + 10 = 0

a = 1, b = 6, c = 10

Discriminant = 6² - 4(1)(10) = 36 - 40 = -4

Negative discriminant → no real solutions (only complex numbers)

But since we’re likely working with real numbers only, answer is “no real solution” or empty set.

✔ Final Answer for #5: ∅ (empty set — no real solutions)

---

Problem 6: -x² - 12x - 18 = 0

Wait — let’s check the original: it says “-x² - 12x - 18 = 0”

But in the student’s work, they wrote:

x = [12 ± √(12² - 4(-1)(-18))] / [2(-1)]

That means they treated it as: a = -1, b = -12, c = -18? Let’s double-check.

Original equation: -x² - 12x - 18 = 0

Multiply both sides by -1 to make it easier: x² + 12x + 18 = 0

Now a = 1, b = 12, c = 18

Discriminant = 12² - 4(1)(18) = 144 - 72 = 72

√72 ≈ 8.485

x = [-12 ± 8.485]/2

First: (-12 + 8.485)/2 = -3.515/2 = -1.7575 ≈ -1.76

Second: (-12 - 8.485)/2 = -20.485/2 = -10.2425 ≈ -10.24

BUT — if we use the original coefficients without multiplying by -1:

Equation: -x² -12x -18 = 0 → a = -1, b = -12, c = -18

Then:

x = [ -(-12) ± √((-12)^2 - 4(-1)(-18)) ] / [2(-1)]

= [12 ± √(144 - 72)] / (-2)

= [12 ± √72] / (-2)

√72 ≈ 8.485

So:

First: (12 + 8.485)/(-2) = 20.485 / (-2) = -10.2425

Second: (12 - 8.485)/(-2) = 3.515 / (-2) = -1.7575

Same answers! So either way works.

✔ Final Answer for #6: {-10.24, -1.76}

---

Final Answer:
{0.20, -10.20}, {5.87, -1.87}, {10, -2}, {9, -4}, ∅, {-10.24, -1.76}