If $x=\frac{2}{3}$ and $x=-3$ are the roots of the equation $ax^2+7x+b=0$, find the values of $a$ and $b$.
Given:
Given equation is $ax^2+7x+b=0$.
To do:
We have to find the values of $a$ and $b$ if $x=\frac{2}{3}$ and $x=-3$ are the roots of the given equation.
Solution:
If $x=m$ is a root of $f(x)$ then $f(m)=0$.
For $x=\frac{2}{3}$
$ax^2+7x+b=0$
$a(\frac{2}{3})^2+7(\frac{2}{3})+b=0$
$\frac{4}{9}a+\frac{14}{3}+b=0$
$9(\frac{4}{9}a)+9(\frac{14}{3})+9(b)=9(0)$ (Multply by $9$ on both sides)
$4a+42+9b=0$
Let it be equation (1).
For $x=-3$
$ax^2+7x+b=0$
$a(-3)^2+7(-3)+b=0$
$9a-21+b=0$
Let it be equation (2).
Solving equations (1) and (2), we get,
$(4a+42+9b=0)-9(9a-21+b=0)$
$4a-81a+42+189+9b-9b=0$
$-77a+231=0$
$77a=231$
$a=\frac{231}{77}$
$a=3$
Substituting $a=3$ in equation (1), we get,
$4(3)+42+9b=0$
$12+42+9b=0$
$9b=-54$
$b=\frac{-54}{9}$
$b=-6$
The values of $a$ and $b$ are $3$ and $-6$ respectively.
Related Articles
- If $x = 0$ and $x = -1$ are the roots of the polynomial $f(x) = 2x^3 - 3x^2 + ax + b$, find the value of $a$ and $b$.
- Find the values of \( a \) and \( b \), if 2 and 3 are zeroes of $x^3+ax^2+bx-30$.
- If roots of equation $x^2+x+1=0$ are $a,\ b$ and roots of $x^2+px+q=0$ are $\frac{a}{b},\ \frac{b}{a}$; then find the value of $p+q$.
- If both $x + 1$ and $x - 1$ are factors of $ax^3 + x^2 - 2x + b$, find the values of $a$ and $b$.
- If $\alpha,\ \beta$ are the roots of the equation $ax^2+bx+c=0$, then find the roots of the equation $ax^2+bx(x+1)+c(x+1)^2=0$.
- If the roots of the equation $(c^2-ab)x^2-2(a^2-bc)x+b^2-ac=0$ are equal, prove that either $a=0$ or $a^3+b^3+c^3=3abc$.
- If $x^3 + ax^2 - bx + 10$ is divisible by $x^2 - 3x + 2$, find the values of $a$ and $b$.
- If \( x-\sqrt{3} \) is a factor of the polynomial \( a x^{2}+b x-3 \) and \( a+b=2-\sqrt{3} \). Find the values of \( a \) and \( b \).
- If $1$ is a root of the quadratic equation $3x^2 + ax - 2 = 0$ and the quadratic equation $a(x^2 + 6x) - b = 0$ has equal roots, find the value of b.
- If $2x + 3$ and $x + 2$ are the factors of the polynomial $g(x) = 2x^3 + ax^2 + 27x + b$, then find the values of the constants a and b in the polynomial g(x).
- If \( 2 x+3 \) and \( x+2 \) are the factors of the polynomial \( g(x)=2 x^{3}+a x^{2}+27 x+b \), find the value of the constants $a$ and $b$.
- Choose the correct answer from the given four options in the following questions:Which of the following equations has the sum of its roots as 3 ?(A) \( 2 x^{2}-3 x+6=0 \)(B) \( -x^{2}+3 x-3=0 \),b>(C) \( \sqrt{2} x^{2}-\frac{3}{\sqrt{2}} x+1=0 \)(D) \( 3 x^{2}-3 x+3=0 \)
- Find the roots of $x\ -\ \frac{3}{x} \ =\ 2$.
- Find the values of $a$ and $b$ so that $(x + 1)$ and $(x - 1)$ are factors of $x^4 + ax^3 - 3x^2 + 2x + b$.
- If the roots of the equation $(a^2+b^2)x^2-2(ac+bd)x+(c^2+d^2)=0$ are equal, prove that $\frac{a}{b}=\frac{c}{d}$.
Kickstart Your Career
Get certified by completing the course
Get Started