Solve By Factoring Worksheet - Fill Online, Printable, Fillable ... - Free Printable

Problem Overview:

The task involves solving quadratic equations by factoring and matching each equation to its correct solution. The goal is to solve each quadratic equation step-by-step using the factoring method and then match the solutions to the provided options.

Quadratic Equation Factoring Method:

1. Write the quadratic equation in standard form: \( ax^2 + bx + c = 0 \).
2. Factor the quadratic expression into two binomials.
3. Use the Zero Product Property: If \( (x - p)(x - q) = 0 \), then \( x = p \) or \( x = q \).
4. Solve for the roots and match them to the given options.

Step-by-Step Solution:

#### 1. \( x^2 - 29x = 0 \)
- Factor out the common term \( x \):
\[
x(x - 29) = 0
\]
- Using the Zero Product Property:
\[
x = 0 \quad \text{or} \quad x - 29 = 0 \implies x = 29
\]
- Solution: \( x = 0 \) or \( x = 29 \)
- Match: Option h

#### 2. \( m^2 + 18m + 81 = 0 \)
- Notice that this is a perfect square trinomial:
\[
m^2 + 18m + 81 = (m + 9)^2
\]
- Set the factor equal to zero:
\[
(m + 9)^2 = 0 \implies m + 9 = 0 \implies m = -9
\]
- Solution: \( m = -9 \)
- Match: Option j

#### 3. \( l^2 + 33l = 0 \)
- Factor out the common term \( l \):
\[
l(l + 33) = 0
\]
- Using the Zero Product Property:
\[
l = 0 \quad \text{or} \quad l + 33 = 0 \implies l = -33
\]
- Solution: \( l = 0 \) or \( l = -33 \)
- Match: Option i

#### 4. \( d^2 + 3q + 70 = 0 \)
- This equation seems to have a typo. Assuming it should be \( d^2 + 3d + 70 = 0 \):
- Try to factor:
\[
d^2 + 3d + 70 = (d + a)(d + b)
\]
where \( a + b = 3 \) and \( ab = 70 \). However, there are no integer factors of 70 that add up to 3. Therefore, this quadratic does not factor nicely over the integers.
- Assuming the problem is correct as written, we cannot solve it using simple factoring. Let's move to the next problem.

#### 5. \( s^2 - 4s = 0 \)
- Factor out the common term \( s \):
\[
s(s - 4) = 0
\]
- Using the Zero Product Property:
\[
s = 0 \quad \text{or} \quad s - 4 = 0 \implies s = 4
\]
- Solution: \( s = 0 \) or \( s = 4 \)
- Match: Option b

#### 6. \( q^2 - 12q + 32 = 0 \)
- Factor the quadratic:
\[
q^2 - 12q + 32 = (q - 4)(q - 8)
\]
- Set each factor equal to zero:
\[
q - 4 = 0 \implies q = 4
\]
\[
q - 8 = 0 \implies q = 8
\]
- Solution: \( q = 4 \) or \( q = 8 \)
- Match: Option c

#### 7. \( h^2 + 67h = 0 \)
- Factor out the common term \( h \):
\[
h(h + 67) = 0
\]
- Using the Zero Product Property:
\[
h = 0 \quad \text{or} \quad h + 67 = 0 \implies h = -67
\]
- Solution: \( h = 0 \) or \( h = -67 \)
- Match: Option g

#### 8. \( p^2 - 16p + 63 = 0 \)
- Factor the quadratic:
\[
p^2 - 16p + 63 = (p - 7)(p - 9)
\]
- Set each factor equal to zero:
\[
p - 7 = 0 \implies p = 7
\]
\[
p - 9 = 0 \implies p = 9
\]
- Solution: \( p = 7 \) or \( p = 9 \)
- Match: Option l

#### 9. \( s^2 - 37s = 0 \)
- Factor out the common term \( s \):
\[
s(s - 37) = 0
\]
- Using the Zero Product Property:
\[
s = 0 \quad \text{or} \quad s - 37 = 0 \implies s = 37
\]
- Solution: \( s = 0 \) or \( s = 37 \)
- Match: Option d

#### 10. \( q^2 + 5q + 4 = 0 \)
- Factor the quadratic:
\[
q^2 + 5q + 4 = (q + 1)(q + 4)
\]
- Set each factor equal to zero:
\[
q + 1 = 0 \implies q = -1
\]
\[
q + 4 = 0 \implies q = -4
\]
- Solution: \( q = -1 \) or \( q = -4 \)
- Match: Option e

Final Matches:

1. \( x^2 - 29x = 0 \) → h
2. \( m^2 + 18m + 81 = 0 \) → j
3. \( l^2 + 33l = 0 \) → i
4. \( d^2 + 3q + 70 = 0 \) → (unsolvable with factoring)
5. \( s^2 - 4s = 0 \) → b
6. \( q^2 - 12q + 32 = 0 \) → c
7. \( h^2 + 67h = 0 \) → g
8. \( p^2 - 16p + 63 = 0 \) → l
9. \( s^2 - 37s = 0 \) → d
10. \( q^2 + 5q + 4 = 0 \) → e

Final Answer:

\[
\boxed{h, j, i, \text{(unsolvable)}, b, c, g, l, d, e}
\]