Solve quadratic equations using the zero product property in this printable math worksheet.

You’ve uploaded a worksheet titled “Zero Product Property”, which asks you to solve each quadratic equation using the Zero Product Property.

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✔ What is the Zero Product Property?

> If the product of two (or more) factors is zero, then at least one of the factors must be zero.

In math:
If \( a \cdot b = 0 \), then \( a = 0 \) or \( b = 0 \).

This property allows us to solve factored equations by setting each factor equal to zero and solving for the variable.

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Let’s solve each problem step-by-step:

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1. \( (x - 1)(x + 4) = 0 \)

Set each factor to zero:

- \( x - 1 = 0 \) → \( x = 1 \)
- \( x + 4 = 0 \) → \( x = -4 \)

✔ Solutions: \( x = 1, -4 \)

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2. \( (m - 5)(m - 2) = 0 \)

Set each factor to zero:

- \( m - 5 = 0 \) → \( m = 5 \)
- \( m - 2 = 0 \) → \( m = 2 \)

✔ Solutions: \( m = 5, 2 \)

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3. \( (z - 7)(z + 2) = 0 \)

Set each factor to zero:

- \( z - 7 = 0 \) → \( z = 7 \)
- \( z + 2 = 0 \) → \( z = -2 \)

✔ Solutions: \( z = 7, -2 \)

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4. \( (9p - 27)(5p + 3) = 0 \)

Set each factor to zero:

- \( 9p - 27 = 0 \) → \( 9p = 27 \) → \( p = 3 \)
- \( 5p + 3 = 0 \) → \( 5p = -3 \) → \( p = -\frac{3}{5} \)

✔ Solutions: \( p = 3, -\frac{3}{5} \)

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5. \( (5r - 9)(4r - 6) = 0 \)

Set each factor to zero:

- \( 5r - 9 = 0 \) → \( 5r = 9 \) → \( r = \frac{9}{5} \)
- \( 4r - 6 = 0 \) → \( 4r = 6 \) → \( r = \frac{6}{4} = \frac{3}{2} \)

✔ Solutions: \( r = \frac{9}{5}, \frac{3}{2} \)

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6. \( (2n + 8)(7n + 25) = 0 \)

Set each factor to zero:

- \( 2n + 8 = 0 \) → \( 2n = -8 \) → \( n = -4 \)
- \( 7n + 25 = 0 \) → \( 7n = -25 \) → \( n = -\frac{25}{7} \)

✔ Solutions: \( n = -4, -\frac{25}{7} \)

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7. \( 8q(q - 3)q = 0 \)

Wait — this is written as \( 8q(q - 3)q \). That’s actually \( 8q^2(q - 3) = 0 \).

So factors are: \( 8 \), \( q^2 \), and \( (q - 3) \). Since 8 ≠ 0, we ignore it.

Set each variable factor to zero:

- \( q^2 = 0 \) → \( q = 0 \)
- \( q - 3 = 0 \) → \( q = 3 \)

✔ Solutions: \( q = 0, 3 \)
*(Note: q=0 is a double root, but we list solutions without repetition unless specified.)*

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8. \( (4t + 5)(2t - 5) = 0 \)

Set each factor to zero:

- \( 4t + 5 = 0 \) → \( 4t = -5 \) → \( t = -\frac{5}{4} \)
- \( 2t - 5 = 0 \) → \( 2t = 5 \) → \( t = \frac{5}{2} \)

✔ Solutions: \( t = -\frac{5}{4}, \frac{5}{2} \)

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9. \( (10v + 5)(11v - 44) = 0 \)

Set each factor to zero:

- \( 10v + 5 = 0 \) → \( 10v = -5 \) → \( v = -\frac{5}{10} = -\frac{1}{2} \)
- \( 11v - 44 = 0 \) → \( 11v = 44 \) → \( v = 4 \)

✔ Solutions: \( v = -\frac{1}{2}, 4 \)

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10. \( (7y + 4)(6y + 1) = 0 \)

Set each factor to zero:

- \( 7y + 4 = 0 \) → \( 7y = -4 \) → \( y = -\frac{4}{7} \)
- \( 6y + 1 = 0 \) → \( 6y = -1 \) → \( y = -\frac{1}{6} \)

✔ Solutions: \( y = -\frac{4}{7}, -\frac{1}{6} \)

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## 📝 Final Answers Summary:

1. \( x = 1, -4 \)
2. \( m = 5, 2 \)
3. \( z = 7, -2 \)
4. \( p = 3, -\frac{3}{5} \)
5. \( r = \frac{9}{5}, \frac{3}{2} \)
6. \( n = -4, -\frac{25}{7} \)
7. \( q = 0, 3 \)
8. \( t = -\frac{5}{4}, \frac{5}{2} \)
9. \( v = -\frac{1}{2}, 4 \)
10. \( y = -\frac{4}{7}, -\frac{1}{6} \)

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✔ You’re all set! These are the solutions using the Zero Product Property. Let me know if you’d like to see how to check any of them!