Question

if the roots of x2+4mx+4m2+m+1=0are then

Answer 1

Correct Question:

If the roots of $x^2+4mx+4m^2+m+1=0$ are equal, then find the value of m.

Final Answer:

When the roots of $x^2+4mx+4m^2+m+1=0$ are equal, then the value of m is -1.

Given:

The roots of $x^2+4mx+4m^2+m+1=0$ are equal

To Find:

When the roots of $x^2+4mx+4m^2+m+1=0$ are equal, then the value of m is to be found out.

Explanation:

The following points are necessary to arrive at the solution.

• The equation having the form $ax^2+bx+c=0$ is a quadratic equation.

• There exist two roots for every quadratic equation.

• The sum of the two roots for the quadratic equation $ax^2+bx+c=0$ is $=-\frac{b}{a}$ .

• The product of the two roots for the quadratic equation $ax^2+bx+c=0$ is $=\frac{c}{a}$.

Step 1 of 3

Assume the two roots of the quadratic equation $x^2+4mx+4m^2+m+1=0$ are $\alpha, \beta$ respectively.

So, write the following two equations.

$\alpha+\beta=-\frac{4m}{1}=-4m \\\alpha \times \beta=\frac{4m^2+m+1}{1} =4m^2+m+1$

Step 2 of 3

Since, the two roots of the equation $x^2+4mx+4m^2+m+1=0$ are equal, the following relation is true.

$\alpha=\beta$

So, these equations are obtained.

$2\alpha=-4m\\\alpha=-2m\\and\\ \alpha^2 =4m^2+m+1$

Step 3 of 3

Now, the value of m is obtained in the following way.

$\alpha^2= \alpha^2\\ (-2m)^2=4m^2+m+1\\4m^2=4m^2+m+1\\m+1=0\\m=-1$

Therefore, the required value of m, where  the roots of the equation $x^2+4mx+4m^2+m+1=0$ are equal, is -1.

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