If 𝑥 equals three is one root of the quadratic equation 𝑥 squared minus two 𝑘𝑥 minus six is equal to zero, find the value of 𝑘.
We’re told that 𝑥 equals three is a root of this quadratic equation. This means that if we substitute 𝑥 equals three into the quadratic expression on the left of the equation, we should get zero. Substituting the value three for 𝑥 gives three squared minus two 𝑘 multiplied by three minus six is equal to zero. And this is an equation that we can solve in order to find the value of 𝑘.
Three squared is nine, and two 𝑘 multiplied by three is six 𝑘. So we have nine minus six 𝑘 minus six is equal to zero. Nine minus six is equal to three, so the equation simplifies to three minus six 𝑘 is equal to zero.
Now to solve this equation, we’ll begin by adding six 𝑘 to each side of the equation as, at the moment, the coefficient or number of 𝑘s that we have is negative and it’s easier to work with a positive number of 𝑘s. Adding six 𝑘 to each side gives three is equal to six 𝑘. To find the value of 𝑘, we need to divide both sides of the equation by six, giving three over six is equal to 𝑘.
Now the fraction three over six can and should be simplified because both the numerator and denominator can be divided by three. Dividing three by three gives one, and dividing six by three gives two. So the value of 𝑘 is one-half.