PLZZ help me with this good explanation with the zero product property and factoring PLZZ read carefully ❤️❤️

In layman's terms, the zero product property is simply a way of saying "anything times zero equals zero".

The x intercepts happen at (0.5, 0), and (-3, 0).

The vertex is a minimum.

The y-intercept happens at (0, -3).

For the last question, graphing f(x) can be done by taking a few key points (the vertex and x-intercepts are very good, as well as the vertex ± 1, 2, etc.).  By that point you have enough data to sketch the curve.  It helps to know in advance what the general shape of the function will be.  In the case of a quadratic function, it will always be a parabola.

Step-by-step explanation:

Let's start by factoring the function:

f(x) = 2x² + 5x - 3

f(x) = 2x² - x + 6x - 3

f(x) = x(2x − 1) + 3 (2x−1)

f(x) = (x + 3)(2x - 1)

Now that we have it factored, finding its x intercept is quite easy.  We just need to see which values of x would would give a y value of zero.  That may sound backward, but note that when y is equal to zero, the graph is touching, or intercepting, the x-axis.

So if we say:

0 = (x + 3)(2x - 1)

then there are two solutions.  First:

0 = (x + 3)(2x - 1)

0 / (x + 3) = (2x - 1)

0 = 2x - 1

2x = 1

x = 0.5

And the other:

0 = (x + 3)(2x - 1)

0 / (2x - 1) = (x + 3)

0 = x + 3

x = -3

So the x intercepts happen at (0.5, 0), and (-3, 0).

Next, regarding whether the vertex is a maximum or minimum, you can find this by the coefficient of the highest power of x in the function.  If the coefficient is negative, then the vertex is a maximum.  If it's positive, then it's a minimum.  In this case the coefficient is 2, so the vertex is a minimum.

For part C, finding the y intercept is easy enough.  Again, counter-intuitively the y intercept happens when x is equal to zero.  So we just need to replace x with zero in the original equation to find it:

f(x) = 2x² + 5x - 3

f(0) = 2(0)² + 5(0) - 3

f(0) = -3

So the y-intercept happens at (0, -3).

2

step-by-step explanation: