Nonlinear equations in 1 variable and systems of equations in 2 variables: Practice Questions

What are the solutions to the equation \( x^2 - 5x + 6 = 0 \)?

How many solutions does the following system of equations have?
\( y = x^2 - 4x + 3 \)
\( y = 2x - 5 \)

What are all the real solutions to the equation \( \sqrt{x + 11} = x - 1 \)?

A system of equations consists of \( y = x^2 - 5 \) and \( y = kx - 9 \). For what value of \( k > 0 \) does the system have exactly one real solution \( (x, y) \)?

The system of equations is given by:
\( y = x^2 - 6x + 10 \)
\( y = x - 2 \)
What is the sum of the \( x \)-coordinates of the two points of intersection of these equations?

A rectangular garden has a length that is 5 meters longer than its width. If the area of the garden is 84 square meters, what is the perimeter of the garden, in meters?

In the \( xy \)-plane, a parabola with equation \( y = x^2 + bx + 9 \) is tangent to the \( x \)-axis at exactly one point. If \( b < 0 \), what is the value of \( b \)?

In the \( xy \)-plane, a system of equations consists of a parabola defined by \( y = x^2 - 6x + 13 \) and a line defined by \( y = 2x + k \), where \( k \) is a constant.
a) Find the value of \( k \) such that the line is tangent to the parabola (the system has exactly one solution).
b) Using the value of \( k \) found in part (a), determine the coordinates \( (x, y) \) of the point of tangency.
c) If \( k = 0 \), use the discriminant of the resulting quadratic equation to determine whether the system has zero, one, or two real solutions. Show the calculation of the discriminant.