Find all possible values of the given variable:
1. \( h^2 + 5h = 0 \)
2. \( z^2 - z = 0 \)
3. \( m^2 + 13m + 40 = 0 \)
4. \( z^2 - 3z = 0 \)
5. \( q^2 + 7q = 0 \)
6. \( k^2 + 2k = 0 \)
7. \( x^2 - 3x - 70 = 0 \)
8. \( q^2 + 7q - 60 = 0 \)
9. \( z^2 + 9z - 36 = 0 \)
10. \( d^2 - 13d + 22 = 0 \)
Answer (2)
1. h 2 + 5 h = 0 h ( x + 5 ) = 0 x = 0 or x + 5 = 0 ∣ − 5 x + 5 − 5 = 0 − 5 x = 0 or x = − 5
2. z 2 − z = 0 z ( x − 1 ) = 0 z = 0 or z − 1 = 0 ∣ + 1 z − 1 + 1 = 0 + 1 x = 0 or z = 1
3. m 2 + 13 m + 40 = 0 a = 1 , b = 13 , c = 40 Δ = b 2 − 4 a c = 1 3 2 − 4 ⋅ 1 ⋅ 40 = 169 − 1600 = − 1431 an d w e kn o w w h e n Δ i s n e g a t i v e , t h eres n o \solution
4. z 2 − 3 z = 0 ( z − 3 ) = 0 z = 0 or z − 3 = 0 ∣ + 3 z − 3 + 3 = 0 + 3 z = 0 or z = 3
5. q 2 + 7 q = 0 q ( q + 7 ) = 0 q = 0 or q + 7 = 0 ∣ − 7 q + 7 − 7 = 0 − 7 q = 0 or q = − 7
6. k 2 + 2 k = 0 k ( k + 2 ) = 0 k = 0 or k + 2 = 0 ∣ − 2 k + 2 − 2 = 0 − 2 k = 0 or k = − 2
7. x 2 − 3 x − 70 = 0 a = 1 , b = − 3 , c = − 70 Δ = b 2 − 4 a c = ( − 3 ) 2 − 4 ⋅ 1 ⋅ ( − 70 ) = 9 + 280 = 289 x 1 = 2 a − b − Δ = 2 3 − 289 = 2 3 − 17 = 2 − 14 = − 7
x 2 = 2 a − b + Δ = 2 3 + 289 = 2 3 + 17 = 2 20 = 10 ( x + 7 ) ( x − 10 ) = 0
8. q 2 + 7 q − 60 = 0 a = 1 , b = 7 , q = − 60 Δ = b 2 − 4 a c = 7 2 − 4 ⋅ 1 ⋅ ( − 60 ) = 49 + 240 = 289 x 1 = 2 a − b − Δ = 2 − 7 − 289 = 2 − 7 − 17 = 2 − 24 = − 12
x 2 = 2 a − b + Δ = 2 − 7 + 289 = 2 − 7 + 17 = 2 10 = 5 ( x + 12 ) ( x − 5 ) = 0
9. z 2 + 9 z − 36 = 0 a = 1 , b = 9 , q = − 36 Δ = b 2 − 4 a c = 9 2 − 4 ⋅ 1 ⋅ ( − 36 ) = 81 + 144 = 225 x 1 = 2 a − b − Δ = 2 − 9 − 225 = 2 − 9 − 15 = 2 − 24 = − 12
x 2 = 2 a − b + Δ = 2 − 9 + 225 = 2 − 9 + 15 = 2 6 = 3 ( x + 11 ) ( x − 3 ) = 0
10. d 2 − 13 d + 22 = 0 a = 1 , b = − 13 , q = 22 Δ = b 2 − 4 a c = ( − 13 ) 2 − 4 ⋅ 1 ⋅ 22 = 169 − 88 = 81 d 1 = 2 a − b − Δ = 2 13 − 81 = 2 13 − 9 = 2 4 = 2
d 2 = 2 a − b + Δ = 2 13 + 81 = 2 13 + 9 = 2 22 = 11 ( d − 2 ) ( d − 11 ) = 0
The solutions to the equations provided are computed through factoring where applicable and applying the quadratic formula. The key results are the roots for each equation including cases with no real solutions. Each equation yields specific values that satisfy the given quadratic forms.
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