A quadratic equation is a polynomial equation of the form ax^2 + bx + c = 0, where a, b, and c are constants and x is the variable. It is called quadratic because the highest power of x is 2.
To find the quadratic equation given one root, you can use the formula (x-r)(x-s) = 0, where r is the known root and s is the other root. In this case, the given root is (2^(1/2)+1), so the other root would be its conjugate (2^(1/2)-1). Plugging these values into the formula, we get (x-(2^(1/2)+1))(x-(2^(1/2)-1)) = 0, which simplifies to x^2 - 2^(1/2)x + 1 = 0. Therefore, the quadratic equation is x^2 - 2^(1/2)x + 1 = 0.
Having a root in the form (2^(1/2)+1) means that the root is irrational, which implies that the quadratic equation has no rational roots. This also means that the graph of the equation will not intersect the x-axis at any rational point.
To graph the quadratic equation given one root, you can plot the known root on the x-axis and use the symmetry property of quadratic functions to plot the other root. Then, you can plot a few additional points by plugging in different values for x and use the shape of the graph to plot the rest of the points.
Yes, a quadratic equation can have two distinct roots, one repeated root, or no real roots. This depends on the discriminant (b^2-4ac) of the equation. If the discriminant is positive, the equation will have two distinct roots. If it is zero, the equation will have one repeated root. If it is negative, the equation will have no real roots.