16.If x= 2/3 and x = – 3 are roots of the quadratic equations ax 2 +7x + b = 0, find the values of a and b.
Answers
Answer:
Answer: The value of a and b is 3 and -6 respectively.
Step-by-step explanation:
Since we have given that
x=\frac{2}{3}\ and\ x=-3x=
3
2
and x=−3 are the roots of the qadratic equation :
ax^2+7x+b=0ax
2
+7x+b=0
First we put the value of x=\frac{2}{3}x=
3
2
in the above quadratic equation:
\begin{lgathered}a(\frac{2}{3})^2+7\times \frac{2}{3}+b=0\\\\\frac{4a}{9}+\frac{14}{3}+b=0\\\\\frac{4a}{9}+b=\frac{-14}{3}\\\\4a+9b=-14\times 3=-42\\\\4a+9b=-42-----------(1)\end{lgathered}
a(
3
2
)
2
+7×
3
2
+b=0
9
4a
+
3
14
+b=0
9
4a
+b=
3
−14
4a+9b=−14×3=−42
4a+9b=−42−−−−−−−−−−−(1)
similarly, if we put the value of x = -3 in the quadratic equation:
\begin{lgathered}-3^2a+7\times -3+b=0\\\\9a-21+b=0\\\\9a+b=21\\\\b=21-9a-------------(2)\end{lgathered}
−3
2
a+7×−3+b=0
9a−21+b=0
9a+b=21
b=21−9a−−−−−−−−−−−−−(2)
From Eq(1) and (2), we get that
\begin{lgathered}4a+9b=-42\\\\4a+9(21-9a)=-42\\\\4a+189-81a=-42\\\\-77a=-42-189\\\\-77a=-231\\\\a=\frac{231}{77}=3\end{lgathered}
4a+9b=−42
4a+9(21−9a)=−42
4a+189−81a=−42
−77a=−42−189
−77a=−231
a=
77
231
=3
Now, we put the value of 'a' in Eq(2), we have
\begin{lgathered}b=21-9a\\\\b=21-9\times 3\\\\b=21-27\\\\b=-6\end{lgathered}
b=21−9a
b=21−9×3
b=21−27
b=−6
Hence, the value of a and b is 3 and -6 respectively.
Step-by-step explanation:
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