Video Transcript
If the roots of the equation 𝑥 squared minus 𝑘𝑥 minus four 𝑘 minus four 𝑥 plus four equals zero are equal, what are the possible values of 𝑘? For each value of 𝑘, work out the roots of the equation.
We will begin by rewriting our equation so it is in the form 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 is equal to zero. Firstly, we have 𝑥 squared minus 𝑘𝑥 minus four 𝑥 minus four 𝑘 plus four equals zero. The second and third terms can be rewritten “plus negative 𝑘 minus four multiplied by 𝑥.” We can rewrite the last two terms as four minus four 𝑘. This gives us the equation 𝑥 squared plus negative 𝑘 minus four 𝑥 plus four minus four 𝑘. This is now written in the general quadratic form 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 equals zero, where 𝑎 is equal to one, the coefficient of 𝑥 squared; 𝑏 is equal to negative 𝑘 minus four, the coefficient of 𝑥; and 𝑐 is equal to four minus four 𝑘, the constant.
We know that if the roots of the quadratic equation are equal, then the discriminant, 𝑏 squared minus four 𝑎𝑐, is equal to zero. Substituting in our values of 𝑎, 𝑏, and 𝑐, we have negative 𝑘 minus four squared minus four multiplied by one multiplied by four minus four 𝑘 is equal to zero. We can expand the first parentheses by multiplying negative 𝑘 minus four by negative 𝑘 minus four. As multiplying two negative terms gives a positive answer, this is equal to 𝑘 squared plus four 𝑘 plus four 𝑘 plus 16.
Collecting like terms gives us 𝑘 squared plus eight 𝑘 plus 16. Negative four multiplied by one is negative four. And we can then distribute this across the parentheses four minus four 𝑘. This gives us negative 16 plus 16𝑘. So our equation becomes 𝑘 squared plus eight 𝑘 plus 16 minus 16 plus 16𝑘 is equal to zero. The 16s cancel, as 16 minus 16 is equal to zero, leaving us with 𝑘 squared plus 24𝑘 is equal to zero.
We can factor out a 𝑘. So we have 𝑘 multiplied by 𝑘 plus 24 equals zero. For the expression on the left-hand side to equal zero, either 𝑘 is equal to zero or 𝑘 plus 24 equals zero. Subtracting 24 from both sides of the second equation gives us 𝑘 is equal to negative 24. The two possible values of 𝑘 are zero and negative 24.
We can now substitute these back in to our original equation to work out the roots. When 𝑘 is equal to zero, we have the quadratic equation 𝑥 squared minus four 𝑥 plus four equals zero. The left-hand side factors to give us 𝑥 minus two multiplied by 𝑥 minus two. The two roots are therefore 𝑥 is equal to two. When 𝑘 is equal to zero, our two roots are 𝑥 equals two and 𝑥 equals two.
We can now repeat this process when 𝑘 is equal to negative 24. This time, our quadratic equation is 𝑥 squared plus 20𝑥 plus 100 is equal to zero. Once again, this can be factored into two equal parentheses, 𝑥 plus 10 multiplied by 𝑥 plus 10. The equal roots of this quadratic equation are 𝑥 is equal to negative 10. Our second possible solutions to the equation when 𝑘 is equal to negative 24 are 𝑥 equals negative 10 and 𝑥 equals negative 10. We have now worked out the two possible values of 𝑘 and the corresponding roots.
