Question

The difference between the roots of the equation 2x^2−5x+c=0 is 0.25. Find c.

Answer 1

Answer:

The value of c is 3.09375.

Step-by-step explanation:

Given : The difference between the roots of the equation $2x^2-5x+c=0$ is 0.25.

To find : The value of c?

Solution :

The general quadratic equation is $ax^2+bx+c=0$ with roots $\alpha,\beta$

The sum of roots is $\alpha+\beta=-\frac{b}{a}$

The product of roots is $\alpha\beta=\frac{c}{a}$

On comparing with given equation, a=2, b=-5 and c=c

Substitute the values,

The sum of roots is $\alpha+\beta=-\frac{-5}{2}$

$\alpha+\beta=\frac{5}{2}$ .....(1)

The product of roots is $\alpha\beta=\frac{c}{2}$ ....(2)

The difference between roots are $\alpha-\beta=0.25$ .....(3)

Using identity,

$\alpha-\beta=\sqrt{(\alpha+\beta)^2-4\alpha\beta}$

Substitute the value in the identity,

$0.25=\sqrt{(\frac{5}{2})^2-4(\frac{c}{2})}$

$0.25=\sqrt{\frac{25}{4}-2c}$

$0.25=\sqrt{\frac{25-8c}{4}}$

$0.25\times 2=\sqrt{25-8c}$

$0.5=\sqrt{25-8c}$

Squaring both side,

$0.5^2=25-8c$

$0.25=25-8c$

$8c=25-0.25$

$8c=24.75$

$c=\frac{24.75}{8}$

$c=3.09375$

Therefore, the value of c is 3.09375.

Answer 2

Answer:

$c=\frac{99}{32}$

Step-by-step explanation:

The given quadratic equation is $2x^2-5x+c=0$.

Comparing this equation to:  $ax^2+bx+c=0$, we have a=2,b=-5.

Where: $x_1+x_2=\frac{5}{2}$ and $x_1x_2=\frac{c}{2}$

The difference in roots is given by:

$x_2-x_1=\sqrt{(x_1+x_2)^2-4x_1x_2}$

$\implies 0.25=\sqrt{(2.5)^2-4(\frac{c}{2})}$

$\implies 0.25^2=6.25-2c$

$\implies 0.0625-6.25=-2c$

$\implies -6.1875=-2c$

Divide both sides by -2

$c=\frac{99}{32}$