Question

Solve the quadratic equations and find the two x- intercepts , and find the y- intercept and sketch the graph of the quadratic equations:y=(x-3)(x+5) and also y=(x+1)(x+6)

Answer 1

To find the x-intercepts, we need to set y equal to zero and solve for x.

For the equation y = (x-3)(x+5):
0 = (x-3)(x+5)
Expanding the equation:
0 = x^2 + 2x - 15
0 = (x+5)(x-3)

Setting each factor to 0:
x+5 = 0 or x-3 = 0
x = -5 or x = 3

So the two x-intercepts are at x = -5 and x = 3.

To find the y-intercept, we set x to 0 in the equation:

y = (0-3)(0+5)
y = -3(5)
y = -15

Therefore, the y-intercept is at y = -15.

For the equation y = (x+1)(x+6):
0 = (x+1)(x+6)
0 = x^2 + 7x + 6

Setting each factor to 0:
x+1 = 0 or x+6 = 0
x = -1 or x = -6

So the two x-intercepts are at x = -1 and x = -6.

To find the y-intercept:
y = (0+1)(0+6)
y = 1*6
y = 6

Therefore, the y-intercept is at y = 6.

Now, we can sketch the graph of each quadratic equation:

For y = (x-3)(x+5), the x-intercepts are (-5,0) and (3,0), and the y-intercept is (0,-15). The graph is a parabola opening upwards passing through these points.

For y = (x+1)(x+6), the x-intercepts are (-1,0) and (-6,0), and the y-intercept is (0,6). The graph is also a parabola opening upwards passing through these points.

I hope this helps! Let me know if you need further assistance.