Question 131760
Write the quadratic equations ({{{ax^2+bx+c = 0}}}) given the solution is x = 8.
Now, quadradic equation have two solutions, so when you say that x = 8 is the only solution, this means the equation has a "double solution" of x = 8 and x = 8.
So, we work backwards from the solutions to the equation as follows:
Start with the solutions:
{{{x = 8}}} and {{{x = 8}}} This means that the parabola generated by the equation touches the x-axis at the point x = 8,
In solving the equation, there had to be two factors that were equal to zero and these factors are:
{{{x-8 = 0}}} and {{{x-8 = 0}}} so if we now multiply these factors together, we will arrive at the original equation.
{{{(x-8)(x-8) = x^2-16x+64 = 0}}} This is the equation.
Check:
We know that if the discriminant ({{{b^2-4ac}}}) of a quadratic equation is equal to zero, then the equation has a double root.
In this case, the equation we just generated shows us that a = 1, b = 16, and c = 64. Let's see if the discriminant does equal zero.
{{{b^2-4ac = 16^2-4(1)(64)}}} = {{{256-256 = 0}}}