Factoring Quadratic Equations Worksheets - Free Printable
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Show Answer Key & Explanations
Step-by-step solution for: Factoring Quadratic Equations Worksheets
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Show Answer Key & Explanations
Step-by-step solution for: Factoring Quadratic Equations Worksheets
To solve the problem of factorizing quadratic equations, we need to express each quadratic equation in the form \((x + p)(x + q) = 0\), where \(p\) and \(q\) are constants that satisfy the conditions:
1. \(p + q = \text{coefficient of } x\)
2. \(pq = \text{constant term}\)
Let's go through each equation step by step.
---
- Coefficient of \(x\): \(3\)
- Constant term: \(2\)
- Find two numbers \(p\) and \(q\) such that:
\[
p + q = 3 \quad \text{and} \quad pq = 2
\]
The numbers are \(p = 1\) and \(q = 2\).
- Factorization:
\[
x^2 + 3x + 2 = (x + 1)(x + 2)
\]
- Solutions:
\[
x = -1 \quad \text{or} \quad x = -2
\]
---
- Coefficient of \(a\): \(7\)
- Constant term: \(6\)
- Find two numbers \(p\) and \(q\) such that:
\[
p + q = 7 \quad \text{and} \quad pq = 6
\]
The numbers are \(p = 1\) and \(q = 6\).
- Factorization:
\[
a^2 + 7a + 6 = (a + 1)(a + 6)
\]
- Solutions:
\[
a = -1 \quad \text{or} \quad a = -6
\]
---
- Coefficient of \(b\): \(4\)
- Constant term: \(4\)
- Find two numbers \(p\) and \(q\) such that:
\[
p + q = 4 \quad \text{and} \quad pq = 4
\]
The numbers are \(p = 2\) and \(q = 2\).
- Factorization:
\[
b^2 + 4b + 4 = (b + 2)(b + 2) = (b + 2)^2
\]
- Solutions:
\[
b = -2 \quad \text{(repeated root)}
\]
---
- Coefficient of \(c\): \(5\)
- Constant term: \(6\)
- Find two numbers \(p\) and \(q\) such that:
\[
p + q = 5 \quad \text{and} \quad pq = 6
\]
The numbers are \(p = 2\) and \(q = 3\).
- Factorization:
\[
c^2 + 5c + 6 = (c + 2)(c + 3)
\]
- Solutions:
\[
c = -2 \quad \text{or} \quad c = -3
\]
---
- Coefficient of \(d\): \(8\)
- Constant term: \(12\)
- Find two numbers \(p\) and \(q\) such that:
\[
p + q = 8 \quad \text{and} \quad pq = 12
\]
The numbers are \(p = 2\) and \(q = 6\).
- Factorization:
\[
d^2 + 8d + 12 = (d + 2)(d + 6)
\]
- Solutions:
\[
d = -2 \quad \text{or} \quad d = -6
\]
---
- Coefficient of \(e\): \(6\)
- Constant term: \(8\)
- Find two numbers \(p\) and \(q\) such that:
\[
p + q = 6 \quad \text{and} \quad pq = 8
\]
The numbers are \(p = 2\) and \(q = 4\).
- Factorization:
\[
e^2 + 6e + 8 = (e + 2)(e + 4)
\]
- Solutions:
\[
e = -2 \quad \text{or} \quad e = -4
\]
---
- Coefficient of \(f\): \(7\)
- Constant term: \(10\)
- Find two numbers \(p\) and \(q\) such that:
\[
p + q = 7 \quad \text{and} \quad pq = 10
\]
The numbers are \(p = 2\) and \(q = 5\).
- Factorization:
\[
f^2 + 7f + 10 = (f + 2)(f + 5)
\]
- Solutions:
\[
f = -2 \quad \text{or} \quad f = -5
\]
---
- Coefficient of \(g\): \(8\)
- Constant term: \(7\)
- Find two numbers \(p\) and \(q\) such that:
\[
p + q = 8 \quad \text{and} \quad pq = 7
\]
The numbers are \(p = 1\) and \(q = 7\).
- Factorization:
\[
g^2 + 8g + 7 = (g + 1)(g + 7)
\]
- Solutions:
\[
g = -1 \quad \text{or} \quad g = -7
\]
---
- Coefficient of \(h\): \(10\)
- Constant term: \(9\)
- Find two numbers \(p\) and \(q\) such that:
\[
p + q = 10 \quad \text{and} \quad pq = 9
\]
The numbers are \(p = 1\) and \(q = 9\).
- Factorization:
\[
h^2 + 10h + 9 = (h + 1)(h + 9)
\]
- Solutions:
\[
h = -1 \quad \text{or} \quad h = -9
\]
---
- Coefficient of \(i\): \(8\)
- Constant term: \(16\)
- Find two numbers \(p\) and \(q\) such that:
\[
p + q = 8 \quad \text{and} \quad pq = 16
\]
The numbers are \(p = 4\) and \(q = 4\).
- Factorization:
\[
i^2 + 8i + 16 = (i + 4)(i + 4) = (i + 4)^2
\]
- Solutions:
\[
i = -4 \quad \text{(repeated root)}
\]
---
- Coefficient of \(j\): \(7\)
- Constant term: \(12\)
- Find two numbers \(p\) and \(q\) such that:
\[
p + q = 7 \quad \text{and} \quad pq = 12
\]
The numbers are \(p = 3\) and \(q = 4\).
- Factorization:
\[
j^2 + 7j + 12 = (j + 3)(j + 4)
\]
- Solutions:
\[
j = -3 \quad \text{or} \quad j = -4
\]
---
- Coefficient of \(k\): \(10\)
- Constant term: \(16\)
- Find two numbers \(p\) and \(q\) such that:
\[
p + q = 10 \quad \text{and} \quad pq = 16
\]
The numbers are \(p = 2\) and \(q = 8\).
- Factorization:
\[
k^2 + 10k + 16 = (k + 2)(k + 8)
\]
- Solutions:
\[
k = -2 \quad \text{or} \quad k = -8
\]
---
- Coefficient of \(m\): \(8\)
- Constant term: \(15\)
- Find two numbers \(p\) and \(q\) such that:
\[
p + q = 8 \quad \text{and} \quad pq = 15
\]
The numbers are \(p = 3\) and \(q = 5\).
- Factorization:
\[
m^2 + 8m + 15 = (m + 3)(m + 5)
\]
- Solutions:
\[
m = -3 \quad \text{or} \quad m = -5
\]
---
- Coefficient of \(n\): \(12\)
- Constant term: \(20\)
- Find two numbers \(p\) and \(q\) such that:
\[
p + q = 12 \quad \text{and} \quad pq = 20
\]
The numbers are \(p = 2\) and \(q = 10\).
- Factorization:
\[
n^2 + 12n + 20 = (n + 2)(n + 10)
\]
- Solutions:
\[
n = -2 \quad \text{or} \quad n = -10
\]
---
- Coefficient of \(p\): \(9\)
- Constant term: \(18\)
- Find two numbers \(p\) and \(q\) such that:
\[
p + q = 9 \quad \text{and} \quad pq = 18
\]
The numbers are \(p = 3\) and \(q = 6\).
- Factorization:
\[
p^2 + 9p + 18 = (p + 3)(p + 6)
\]
- Solutions:
\[
p = -3 \quad \text{or} \quad p = -6
\]
---
- Coefficient of \(q\): \(12\)
- Constant term: \(32\)
- Find two numbers \(p\) and \(q\) such that:
\[
p + q = 12 \quad \text{and} \quad pq = 32
\]
The numbers are \(p = 4\) and \(q = 8\).
- Factorization:
\[
q^2 + 12q + 32 = (q + 4)(q + 8)
\]
- Solutions:
\[
q = -4 \quad \text{or} \quad q = -8
\]
---
\[
\boxed{
\begin{array}{|c|c|c|}
\hline
1 & x^2 + 3x + 2 = 0 & (x + 1)(x + 2) \\
2 & a^2 + 7a + 6 = 0 & (a + 1)(a + 6) \\
3 & b^2 + 4b + 4 = 0 & (b + 2)^2 \\
4 & c^2 + 5c + 6 = 0 & (c + 2)(c + 3) \\
5 & d^2 + 8d + 12 = 0 & (d + 2)(d + 6) \\
6 & e^2 + 6e + 8 = 0 & (e + 2)(e + 4) \\
7 & f^2 + 7f + 10 = 0 & (f + 2)(f + 5) \\
8 & g^2 + 8g + 7 = 0 & (g + 1)(g + 7) \\
9 & h^2 + 10h + 9 = 0 & (h + 1)(h + 9) \\
10 & i^2 + 8i + 16 = 0 & (i + 4)^2 \\
11 & j^2 + 7j + 12 = 0 & (j + 3)(j + 4) \\
12 & k^2 + 10k + 16 = 0 & (k + 2)(k + 8) \\
13 & m^2 + 8m + 15 = 0 & (m + 3)(m + 5) \\
14 & n^2 + 12n + 20 = 0 & (n + 2)(n + 10) \\
15 & p^2 + 9p + 18 = 0 & (p + 3)(p + 6) \\
16 & q^2 + 12q + 32 = 0 & (q + 4)(q + 8) \\
\hline
\end{array}
}
\]
1. \(p + q = \text{coefficient of } x\)
2. \(pq = \text{constant term}\)
Let's go through each equation step by step.
---
1. \(x^2 + 3x + 2 = 0\)
- Coefficient of \(x\): \(3\)
- Constant term: \(2\)
- Find two numbers \(p\) and \(q\) such that:
\[
p + q = 3 \quad \text{and} \quad pq = 2
\]
The numbers are \(p = 1\) and \(q = 2\).
- Factorization:
\[
x^2 + 3x + 2 = (x + 1)(x + 2)
\]
- Solutions:
\[
x = -1 \quad \text{or} \quad x = -2
\]
---
2. \(a^2 + 7a + 6 = 0\)
- Coefficient of \(a\): \(7\)
- Constant term: \(6\)
- Find two numbers \(p\) and \(q\) such that:
\[
p + q = 7 \quad \text{and} \quad pq = 6
\]
The numbers are \(p = 1\) and \(q = 6\).
- Factorization:
\[
a^2 + 7a + 6 = (a + 1)(a + 6)
\]
- Solutions:
\[
a = -1 \quad \text{or} \quad a = -6
\]
---
3. \(b^2 + 4b + 4 = 0\)
- Coefficient of \(b\): \(4\)
- Constant term: \(4\)
- Find two numbers \(p\) and \(q\) such that:
\[
p + q = 4 \quad \text{and} \quad pq = 4
\]
The numbers are \(p = 2\) and \(q = 2\).
- Factorization:
\[
b^2 + 4b + 4 = (b + 2)(b + 2) = (b + 2)^2
\]
- Solutions:
\[
b = -2 \quad \text{(repeated root)}
\]
---
4. \(c^2 + 5c + 6 = 0\)
- Coefficient of \(c\): \(5\)
- Constant term: \(6\)
- Find two numbers \(p\) and \(q\) such that:
\[
p + q = 5 \quad \text{and} \quad pq = 6
\]
The numbers are \(p = 2\) and \(q = 3\).
- Factorization:
\[
c^2 + 5c + 6 = (c + 2)(c + 3)
\]
- Solutions:
\[
c = -2 \quad \text{or} \quad c = -3
\]
---
5. \(d^2 + 8d + 12 = 0\)
- Coefficient of \(d\): \(8\)
- Constant term: \(12\)
- Find two numbers \(p\) and \(q\) such that:
\[
p + q = 8 \quad \text{and} \quad pq = 12
\]
The numbers are \(p = 2\) and \(q = 6\).
- Factorization:
\[
d^2 + 8d + 12 = (d + 2)(d + 6)
\]
- Solutions:
\[
d = -2 \quad \text{or} \quad d = -6
\]
---
6. \(e^2 + 6e + 8 = 0\)
- Coefficient of \(e\): \(6\)
- Constant term: \(8\)
- Find two numbers \(p\) and \(q\) such that:
\[
p + q = 6 \quad \text{and} \quad pq = 8
\]
The numbers are \(p = 2\) and \(q = 4\).
- Factorization:
\[
e^2 + 6e + 8 = (e + 2)(e + 4)
\]
- Solutions:
\[
e = -2 \quad \text{or} \quad e = -4
\]
---
7. \(f^2 + 7f + 10 = 0\)
- Coefficient of \(f\): \(7\)
- Constant term: \(10\)
- Find two numbers \(p\) and \(q\) such that:
\[
p + q = 7 \quad \text{and} \quad pq = 10
\]
The numbers are \(p = 2\) and \(q = 5\).
- Factorization:
\[
f^2 + 7f + 10 = (f + 2)(f + 5)
\]
- Solutions:
\[
f = -2 \quad \text{or} \quad f = -5
\]
---
8. \(g^2 + 8g + 7 = 0\)
- Coefficient of \(g\): \(8\)
- Constant term: \(7\)
- Find two numbers \(p\) and \(q\) such that:
\[
p + q = 8 \quad \text{and} \quad pq = 7
\]
The numbers are \(p = 1\) and \(q = 7\).
- Factorization:
\[
g^2 + 8g + 7 = (g + 1)(g + 7)
\]
- Solutions:
\[
g = -1 \quad \text{or} \quad g = -7
\]
---
9. \(h^2 + 10h + 9 = 0\)
- Coefficient of \(h\): \(10\)
- Constant term: \(9\)
- Find two numbers \(p\) and \(q\) such that:
\[
p + q = 10 \quad \text{and} \quad pq = 9
\]
The numbers are \(p = 1\) and \(q = 9\).
- Factorization:
\[
h^2 + 10h + 9 = (h + 1)(h + 9)
\]
- Solutions:
\[
h = -1 \quad \text{or} \quad h = -9
\]
---
10. \(i^2 + 8i + 16 = 0\)
- Coefficient of \(i\): \(8\)
- Constant term: \(16\)
- Find two numbers \(p\) and \(q\) such that:
\[
p + q = 8 \quad \text{and} \quad pq = 16
\]
The numbers are \(p = 4\) and \(q = 4\).
- Factorization:
\[
i^2 + 8i + 16 = (i + 4)(i + 4) = (i + 4)^2
\]
- Solutions:
\[
i = -4 \quad \text{(repeated root)}
\]
---
11. \(j^2 + 7j + 12 = 0\)
- Coefficient of \(j\): \(7\)
- Constant term: \(12\)
- Find two numbers \(p\) and \(q\) such that:
\[
p + q = 7 \quad \text{and} \quad pq = 12
\]
The numbers are \(p = 3\) and \(q = 4\).
- Factorization:
\[
j^2 + 7j + 12 = (j + 3)(j + 4)
\]
- Solutions:
\[
j = -3 \quad \text{or} \quad j = -4
\]
---
12. \(k^2 + 10k + 16 = 0\)
- Coefficient of \(k\): \(10\)
- Constant term: \(16\)
- Find two numbers \(p\) and \(q\) such that:
\[
p + q = 10 \quad \text{and} \quad pq = 16
\]
The numbers are \(p = 2\) and \(q = 8\).
- Factorization:
\[
k^2 + 10k + 16 = (k + 2)(k + 8)
\]
- Solutions:
\[
k = -2 \quad \text{or} \quad k = -8
\]
---
13. \(m^2 + 8m + 15 = 0\)
- Coefficient of \(m\): \(8\)
- Constant term: \(15\)
- Find two numbers \(p\) and \(q\) such that:
\[
p + q = 8 \quad \text{and} \quad pq = 15
\]
The numbers are \(p = 3\) and \(q = 5\).
- Factorization:
\[
m^2 + 8m + 15 = (m + 3)(m + 5)
\]
- Solutions:
\[
m = -3 \quad \text{or} \quad m = -5
\]
---
14. \(n^2 + 12n + 20 = 0\)
- Coefficient of \(n\): \(12\)
- Constant term: \(20\)
- Find two numbers \(p\) and \(q\) such that:
\[
p + q = 12 \quad \text{and} \quad pq = 20
\]
The numbers are \(p = 2\) and \(q = 10\).
- Factorization:
\[
n^2 + 12n + 20 = (n + 2)(n + 10)
\]
- Solutions:
\[
n = -2 \quad \text{or} \quad n = -10
\]
---
15. \(p^2 + 9p + 18 = 0\)
- Coefficient of \(p\): \(9\)
- Constant term: \(18\)
- Find two numbers \(p\) and \(q\) such that:
\[
p + q = 9 \quad \text{and} \quad pq = 18
\]
The numbers are \(p = 3\) and \(q = 6\).
- Factorization:
\[
p^2 + 9p + 18 = (p + 3)(p + 6)
\]
- Solutions:
\[
p = -3 \quad \text{or} \quad p = -6
\]
---
16. \(q^2 + 12q + 32 = 0\)
- Coefficient of \(q\): \(12\)
- Constant term: \(32\)
- Find two numbers \(p\) and \(q\) such that:
\[
p + q = 12 \quad \text{and} \quad pq = 32
\]
The numbers are \(p = 4\) and \(q = 8\).
- Factorization:
\[
q^2 + 12q + 32 = (q + 4)(q + 8)
\]
- Solutions:
\[
q = -4 \quad \text{or} \quad q = -8
\]
---
Final Answer
\[
\boxed{
\begin{array}{|c|c|c|}
\hline
1 & x^2 + 3x + 2 = 0 & (x + 1)(x + 2) \\
2 & a^2 + 7a + 6 = 0 & (a + 1)(a + 6) \\
3 & b^2 + 4b + 4 = 0 & (b + 2)^2 \\
4 & c^2 + 5c + 6 = 0 & (c + 2)(c + 3) \\
5 & d^2 + 8d + 12 = 0 & (d + 2)(d + 6) \\
6 & e^2 + 6e + 8 = 0 & (e + 2)(e + 4) \\
7 & f^2 + 7f + 10 = 0 & (f + 2)(f + 5) \\
8 & g^2 + 8g + 7 = 0 & (g + 1)(g + 7) \\
9 & h^2 + 10h + 9 = 0 & (h + 1)(h + 9) \\
10 & i^2 + 8i + 16 = 0 & (i + 4)^2 \\
11 & j^2 + 7j + 12 = 0 & (j + 3)(j + 4) \\
12 & k^2 + 10k + 16 = 0 & (k + 2)(k + 8) \\
13 & m^2 + 8m + 15 = 0 & (m + 3)(m + 5) \\
14 & n^2 + 12n + 20 = 0 & (n + 2)(n + 10) \\
15 & p^2 + 9p + 18 = 0 & (p + 3)(p + 6) \\
16 & q^2 + 12q + 32 = 0 & (q + 4)(q + 8) \\
\hline
\end{array}
}
\]
Parent Tip: Review the logic above to help your child master the concept of solve quadratic equation by factoring worksheet.
