RTCNTC said:Find the value of k such that the equation has exactly one root.
3x^2 + sqrt{2k}x + 6 = 0
This question involves the discriminant, right?
b^2 - 4ac = 0
(sqrt{2k}^2 - 4(3)(6) = 0
2k - 72 = 0
2k = 72
k = 72/2
k = 36
Correct?
greg1313 said:Alternatively,
$$3x^2+\sqrt{2k}x+6=0$$
$$3\left(x^2+\frac{\sqrt{2k}}{3}x\right)+6=0$$
$$3\left(x+\frac{\sqrt{2k}}{6}\right)^2+6-3\left(\frac{\sqrt{2k}}{6}\right)^2=0$$
If the equation has exactly one real root then the vertex "touches" (is tangent to) the $x$-axis, so
$$6-3\left(\frac{\sqrt{2k}}{6}\right)^2=0$$
$$6=3\left(\frac{\sqrt{2k}}{6}\right)^2$$
$$6=3\frac{2k}{36}$$
$$2=\frac{2k}{36}$$
$$72=2k$$
$$k=36$$
The value of k in this equation helps determine the solutions for x, which are the points where the equation crosses the x-axis. It is a constant that affects the shape and position of the parabola.
To solve for k, you can use the quadratic formula or complete the square. You will need to substitute the given values of x and solve for k using algebraic manipulation.
The coefficient of x^2, which is 3 in this equation, determines the direction and steepness of the parabola. It also affects the position of the vertex, which can give clues about the value of k.
Yes, there can be multiple values of k that satisfy this equation. This is because there can be multiple points where the parabola crosses the x-axis, each with a different x value. Each of these points corresponds to a different value of k.
Finding k is important in various fields such as physics, engineering, and economics. It can help determine the optimal value for a variable in a given situation, and also provide insights into the behavior of a system.