Question

Find the condition that the equations ax^2 + bx + c = 0 and a1x^2 + b1x + c1 = 0 may
have a common root. Find this common root, when it exits​

Answer 1

Step-by-step explanation:

Let the two quadratic equations are a1x^2 + b1x + c1 = 0 and a2x^2 + b2x + c2 = 0

Now we are going to find the condition that the above quadratic equations may have a common root.

Let α be the common root of the equations a1x^2 + b1x + c1 = 0 and a2x^2 + b2x + c2 = 0. Then,

a1α^2 + b1α + c1 = 0

a2α^2 + b2α + c2 = 0

Now, solving the equations a1α^2 + b1α + c1 = 0, a2α^2 + b2α + c2 = 0 by cross-multiplication, we get

α^2/b1c2 - b2c1 = α/c1a2 - c2a1 = 1/a1b2 - a2b1

⇒ α = b1c2 - b2c1/c1a2 - c2a1, (From first two)

Or, α = c1a2 - c2a1/a1b2 - a2 b1, (From 2nd and 3rd)

⇒ b1c2 - b2c1/c1a2 - c2a1 = c1a2 - c2a1/a1b2 - a2b1

⇒ (c1a2 - c2a1)^2 = (b1c2 - b2c1)(a1b2 - a2b1), which is the required condition for one root to be common of two quadratic equations.

The common root is given by α = c1a2 - c2a1/a1b2 - a2b1 or, α = b1c2 - b2c1/c1q2 - c2a1

Note: (i) We can find the common root by making the same coefficient of x^2 of the given equations and then subtracting the two equations.

(ii) We can find the other root or roots by using the relations between roots and coefficients of the given equations

Condition for both roots common:

Let α, β be the common roots of the quadratic equations a1x^2 + b1x + c1 = 0 and a2x^2 + b2x + c2 = 0. Then

α + β = -b1/a1, αβ = c1/a1 and α + β = -b2/a2, αβ = c2/a2

Therefore, -b/a1 = - b2/a2 and c1/a1 = c2/a2

⇒ a1/a2 = b1/b2 and a1/a2 = c1/c2

⇒ a1/a2 = b1/b2 = c1/c2

This is the required condition.

Answer 2

Answer:

Final Answer.

Step-by-step explanation:

Given,

Quadratic Equations:

$ax^{2}$ + bx + c = 0, and

$a_{1}$$x^{2}$ + $b_{1}$x + $c_{1}$ = 0

Cross multiplying these two equations, we get

$a^{2}$/(b$c_{1}$ - $b_{1}$c) = a/(c$a_{1}$ - $c_{1}$a) = 1/(a$b_{1}$ - $a_{1}$b)

From the first two equalities,

⇒ a = (b$c_{1}$ - $b_{1}$c)/(c$a_{1}$ - $c_{1}$a)

We can also get a from the other two equalities,

a = (c$a_{1}$ - $c_{1}$a)/(a$b_{1}$ - $a_{1}$b)

⇒  (b$c_{1}$ - $b_{1}$c)/(c$a_{1}$ - $c_{1}$a) = (c$a_{1}$ - $c_{1}$a)/(a$b_{1}$ - $a_{1}$b)

The above equation is the condition for common roots of two quadratic equation.

Now, let $\alpha$ and $\beta$ be the common roots of the equations given

Then,

$\alpha$ + $\beta$ = $\frac{-b}{a}$

$\alpha$.$\beta$ = $\frac{c}{a}$

and,

$\alpha$ + $\beta$ = $-b_{1}$/$a_{1}$

$\alpha$.$\beta$ = $c_{1}$/$a_{1}$

Therefore,

$\frac{-b}{a}$ = $-b_{1}$/$a_{1}$

and,

$\frac{c}{a}$ = $c_{1}$/$a_{1}$

To find, common roots if there exists any

⇒ a/$a_{1}$ = b/$b_{1}$

and,

a/$a_{1}$ = c/$c_{1}$

Therefore, the common roots are:

a/$a_{1}$ = b/$b_{1}$ = c/$c_{1}$

Hence, the final solution.

#SPJ2