Find the value of equation with equal roots

• Nov 16th 2010, 06:56 AM
watp
Find the value of equation with equal roots
I'm trying to solve this equation, but have no clue how I'm supposed to do it. My teacher tends to give out homework much more difficult than we've done in class.

Find the value of k which results in the equation having equal roots, given that k <> 0
\$\displaystyle kx^2 + 2kx - 1 = 0\$

• Nov 16th 2010, 07:03 AM
Unknown008
Use the fact that if the equation in the form \$\displaystyle ax^2 + bx + c\$ has equal roots, then you have:

\$\displaystyle b^2 - 4ac = 0\$

Sub in and solve for k.
• Nov 16th 2010, 07:13 AM
watp
Thank you Unknown!
• Nov 16th 2010, 07:14 AM
Unknown008
You're welcome (Smile)
• Nov 16th 2010, 07:26 AM
emakarov
Yes. Here is another way. The polynomial \$\displaystyle x^2+2x-1/k\$ has roots iff it can be factored as \$\displaystyle (x-a_1)(x-a_2)\$ for some \$\displaystyle a_1, a_2\$. Here \$\displaystyle a_1, a_2\$ are, in fact, the roots. The right-to-left direction is obvious. One way to show the left-to-right direction is to apply the little Bézout's theorem. So if we are looking for the situation when both roots are \$\displaystyle a\$, then \$\displaystyle (x-a)^2=x^2-2ax+a^2=x^2+2x-1/k\$. Now, polynomials are equal iff their corresponding coefficients are equal, so \$\displaystyle -2a=2\$ and \$\displaystyle a^2=-1/k\$. From here, one can find \$\displaystyle k\$.

A similar way is to use Viète's formulas, which says that if \$\displaystyle a_1, a_2\$ are roots of \$\displaystyle x^2+2x-1/k\$, then \$\displaystyle a_1+a_2=-2\$ and \$\displaystyle a_1a_2=-1/k\$.
• Nov 17th 2010, 02:03 AM
HallsofIvy
And, just to stick in my oar, although this is serious overkill, a function, f(x), has a double zero at x= a if and only if both f(a)= 0 and f'(a)= 0. Solve the two equations \$\displaystyle kx^2+ 2kx- 1= 0\$ and \$\displaystyle 2kx+ 2k= 0\$.