Question

If the roots of the equation x^2 + 14x - m = 0 are in the ratio 4:3, then find the value of m. ?​

Answer 1

Step-by-step explanation:

Given :-

The roots of the equation x^2+ 14x - m = 0 are in the ratio 4:3

To find :-

Find the value of m. ?

Solution :-

Given equation is x^2 + 14x - m = 0

On comparing with the standard quadratic equation ax^2+bx+c = 0

a = 1

b = 14

c = -m

Ratio of the roots of the equation = 4:3

Let they be 4a and 3a

We know that

Sum of the roots = -b/a

=> 4a+3a = -14/1

=> 7a = -14

=> a = -14/7

=> a = -2

Therefore, Value of a = -2

and

Product of the roots = c/a

=> (4a)(3a) = -m/1

=> 12a^2 = -m

=> 12(-2)^2 = -m

=> 12(4) = -m

=> 48 = -m

=> -m = 48

=> m = -48

Therefore, m = -48

Answer:-

The value of m for the given problem is -48

Used formulae:-

→ The standard quadratic equation ax^2+bx+c = 0

→ Sum of the roots = -b/a

→ Product of the roots = c/a

Answer 2

Given:

The roots of the equation$\[{x^2} + 14x - m = 0\]$are in the ratio $4:3$.

To find : Find the value of m?

Solution :-

Given equation is $\[{x^2} + 14x - m = 0\]$

Compare the given equation with the standard quadratic equation $\[a{x^2} + bx + c = 0\]$

$\[\begin{array}{*{20}{l}}{a = 1}\\{b = 14}\\{c = - m}\end{array}\]$  

Understand that, Ratio of the roots of the equation $= 4:3$

Let Ratio of the roots of the equation be 4a and 3a respectively.

We know that

Sum of the roots$\[ = - \frac{b}{a}\]$  

$\[\begin{array}{l} \Rightarrow 4a + 3a = - \frac{{14}}{1}\\ \Rightarrow 7a = - 14\\ \Rightarrow a = - 2\end{array}\]$

Therefore, Value of $a = -2$

Find the value of m.

Product of the roots$\[ = \frac{c}{a}\]$  

$\[\begin{array}{l} \Rightarrow \left( {4a} \right)\left( {3a} \right) = - \frac{m}{1}\\ \Rightarrow 12{a^2} = - m\\ \Rightarrow 12{\left( { - 2} \right)^2} = - m\\ \Rightarrow 12 \times 4 = - m\\ \Rightarrow 48 = - m\\ \Rightarrow m = - 48\end{array}\]$

Therefore, $m = -48$

Hence, The value of m for the given problem is $-48$.