Grade 11Algebra
If x2-hx-21=0,x2-3hx+35=0(h>0)has a common root ,then value of h is equal to A 1. B 2. C 3. D 4.
If x2-hx-21=0,x2-3hx+35=0(h>0)has a common root ,then value of h is equal to
A 1. B 2. C 3. D 4.
1 Answer
Arun
► Let 𝑎 be the common roots of:
𝑥² – ℎ𝑥 – 21 = 0
and
𝑥² – 3ℎ𝑥 + 35 = 0
Then by definition of a root, we have the system:
{ 𝑎² – ℎ𝑎 – 21 = 0
{ 𝑎² – 3ℎ𝑎 + 35 = 0
► Subtracting second equation from the first yields:
(𝑎² – ℎ𝑎 – 21) – (𝑎² – 3ℎ𝑎 + 35) = 0 – 0
𝑎² – ℎ𝑎 – 21 – 𝑎² + 3ℎ𝑎 – 35 = 0
𝑎² – 𝑎² – ℎ𝑎 + 3ℎ𝑎 – 21 – 35 = 0
0 + 2ℎ𝑎 – 56 = 0
2ℎ𝑎 = 56
𝑎 = 28/ℎ
► Inserting this in the first equation:
𝑎² – ℎ𝑎 – 21 = 0
(28/ℎ)² – ℎ(28/ℎ) – 21 = 0 ← Since 𝑎=28/ℎ
784/ℎ² – 28 – 21 = 0
784/ℎ² – 49 = 0
784/ℎ² = 49
16/ℎ² = 1 ← Divide both sides by 49
16 = ℎ² ← Multiply both sides by ℎ²
ℎ = ±4
► Now since we are told that ℎ>0, the solution ℎ=-4 is discarded leading to the final conclusion that:
ℎ = 4 ◄ANSWER
𝑥² – ℎ𝑥 – 21 = 0
and
𝑥² – 3ℎ𝑥 + 35 = 0
Then by definition of a root, we have the system:
{ 𝑎² – ℎ𝑎 – 21 = 0
{ 𝑎² – 3ℎ𝑎 + 35 = 0
► Subtracting second equation from the first yields:
(𝑎² – ℎ𝑎 – 21) – (𝑎² – 3ℎ𝑎 + 35) = 0 – 0
𝑎² – ℎ𝑎 – 21 – 𝑎² + 3ℎ𝑎 – 35 = 0
𝑎² – 𝑎² – ℎ𝑎 + 3ℎ𝑎 – 21 – 35 = 0
0 + 2ℎ𝑎 – 56 = 0
2ℎ𝑎 = 56
𝑎 = 28/ℎ
► Inserting this in the first equation:
𝑎² – ℎ𝑎 – 21 = 0
(28/ℎ)² – ℎ(28/ℎ) – 21 = 0 ← Since 𝑎=28/ℎ
784/ℎ² – 28 – 21 = 0
784/ℎ² – 49 = 0
784/ℎ² = 49
16/ℎ² = 1 ← Divide both sides by 49
16 = ℎ² ← Multiply both sides by ℎ²
ℎ = ±4
► Now since we are told that ℎ>0, the solution ℎ=-4 is discarded leading to the final conclusion that:
ℎ = 4 ◄ANSWER
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