MHB Solve Algebra Quadratics: Get Solutions & Answers

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Hi all,

Son came home with these questions and is unable to solve them.

I am also struggling with these questions. I need to explain to my son how to get the solutions and not just the answers.

Any help would be greatly appreciated!

1. Two cars are traveling along two straight roads which are perpendicular to each other and meet at the point O.
The first car starts 50km west of O and travels east at a constant speed of 20km/h.
The second car starts 30km south of O at the same time and travels north at a constant speed of 15km/h.

a)Show that at time t, the distance d between the two cars satisfies:

d^2 = 625t^2 - 2900t + 3400

B) Hence find the closest distance between the two cars.2. a) The graph of y =x^2 - 6x + k has its vertex on the x axis. Find the value of k.

b) A second parabola has its vertex at (-2, 5) and passes through the vertex of the first graph. Find the equation of the second graph in the form y = ax^2 + bx + c

c) Find the co-ordinates of the other point of intersection between the two graphs.Many thanks!

 

[Evgeny.Makarov]

1. Two cars are traveling along two straight roads which are perpendicular to each other and meet at the point O.
The first car starts 50km west of O and travels east at a constant speed of 20km/h.
The second car starts 30km south of O at the same time and travels north at a constant speed of 15km/h.

a)Show that at time t, the distance d between the two cars satisfies:

d^2 = 625t^2 - 2900t + 3400

Express the distance of each car to the intersection point as a function of $t$, then use the Pythagorean theorem, which expresses the hypotenuse through sides: $c^2=a^2+b^2$.

B) Hence find the closest distance between the two cars.

The vertex of the parabola $ax^2+bx+c$ has $x$ coordinate $-\frac{b}{2a}$. Calculate the value of the parabola at this point.

2. a) The graph of y =x^2 - 6x + k has its vertex on the x axis. Find the value of k.

Again, find the $x$ coordinate of the vertex, substitute it into $x^2 - 6x + k$ and equate to 0 (since the vertex has $y$ coordinate equal to 0).

b) A second parabola has its vertex at (-2, 5) and passes through the vertex of the first graph. Find the equation of the second graph in the form y = ax^2 + bx + c

All parabolas with vertex $(x_0,y_0)$ have equation $a(x-x_0)^2+y_0$ for some $a$. Use the fact that the parabola passes through the vertex of $y =x^2 - 6x + k$ you found in the previous question to find $a$.