find the value of k for which the equation 3x^2+10x+k=0 has equal roots
You need to remember that the quadratic formula of a quadratic equation tells the number and the values of roots of equation.
You need to remember the standard form of the quadratic equation such that:
`ax^2 + bx + c = 0`
You need to remember the quadratic formula that helps you to find the roots such that:
`x_(1,2) = (-b+-sqrt(b^2-4ac))/(2a)`
Notice that the expression under the square root needs to be positive or at most zero for the square root to be posssible.
If the expression under the square root is strictly positive, then the square root gives you two values`:+-sqrt(b^2-4ac).`
`` If the expression under the square root is equal to zero, then the square root gives one value, hence `x_1 = (-b+0)/(2a) = (-b-0)/(2a) = x_2` .
Notice that if `b^2-4ac = 0` , the values of roots are equal, hence you need to solve equation `b^2-4ac = 0` .
Substituting 3,10,k for a,b,c yields:
`100 - 12k = 0`
You need to solve for k this equation such that:
`-12k = -100 =gt k = 100/12 =gt k = 50/6 =gt k = 25/3`
Hence, the quadratic equation have two equal roots if the coefficient `k = 25/3` .
The value of k in the quadratic equation 3x^2+10x+k=0 has to be determined so that the roots are equal.
For the roots of a quadratic equation ax^2 + bx + c = 0 to be equal b^2 = 4ac.
Substituting the values in the equation that is given
10^2 = 4*k*3
=> 100 = 12k
=> k = 100/12
=> k = 25/3
The required value of k = 25/3