help please!

Question

Sketch the graph of the quadratic function Y-= 3x^2 + 5x + 3 on grid. How many solutions are there for the quadratic equation 3x^2 + 5x + 3 = 0. Explain how you know this by looking at the graph.

Answer 1

The quadratic function is y = 3x² + 5x + 3.

The related quadratic equation is 3x² + 5x + 3 = 0.

3x² + 5x + 3 = 0

Completing the square Method :

The equation is 3x² + 5x + 3 = 0.

Separate variables and constants aside by subtracting 3 from each side.

3x² + 5x + 3 - 3 = 0 - 3

3x² + 5x = - 3

Divide each side by 3.

3x²/3 + 5x/3 = - 3/3

x² + 5x/3 = - 1

To change the expression into a perfect square trinomial add (half the x coefficient)² to each side of the expression.

 Here x coefficient = 5/3. So, (half the x coefficient)² = (5/6)²= 25/36.

Add 25/36 to each side.

x² + 5x/3 + 25/36 = - 1 + 25/36

(x + 5/6)² = - 11/36

x + 5/6 = ± √(11/36)

x = - 5/6 ± √11/6.

The solutions are x = (- 5 + √11)/6 and x = (- 5 - √11)/6.

Graph :

Make the table of values to find ordered pairs that satisfy the equation.

Choose values for x and find the corresponding values for y.

x	y = 3x² + 5x + 3	(x, y)
-3	y = 3(-3)² + 5(-3) + 3 = 27 - 15 + 3 = 15	(-3, 15)
-2	y = 3(-2)² + 5(-2) + 3 = 12 - 10 + 3 = 5	(-2, 5)
-1	y = 3(-1)² + 5(-1) + 3 = 3 - 5 + 3 = 1	(-1, 1)
0	y = 3(0)² + 5(0) + 3 = 0 + 0 + 3 = 3	(0, 3)
1	y = 3(1)² + 5(1) + 3 = 3 + 5 + 3 = 11	(1, 11)

1.Draw a coordinate plane.

2.Plot the coordinate points.

3.Then sketch the graph, connecting the points with a smooth curve.