In the next several units, we will be looking at quadratic equations. A **quadratic equation **is just an equation that has an (x^2) and the highest exponent is 2. When we graph a quadratic equation, the shape of the graph is a **parabola**. A parabola is basically just a right-side-up or upside-down “U”- shape. The highest or lowest point on the parabola is the **vertex. **A vertical line of symmetry runs through the vertex.

The **standard form** of a parabola is: (y=ax^2+bx+c)

The *a *value determines how wide the parabola is and which direction the parabola opens. If *a *is positive, the parabola opens upward. If *a *is negative, the parabola opens downward. The *c *represents the *y*-intercept.

Here are two examples of parabolas:

(y=2x^2-4x-6)

- The
*a*value is positive, so the graph opens upward. - The
*c*value is –6. This tells us the*y*-intercept.

(y=-x^2-x+6)

- The
*a*value is negative, so the graph opens downward. - The
*c*value is 6. This tells us the*y*-intercept.

We can factor the two equations above to find the *x*-intercepts (aka the zeros or solutions). To factor the equation on the left, we would first divide each term by 2 to get: (y=x^2-2x-3). When we factor, we want to figure out what two binomial terms we could multiply together to get our equation. We’ll look a little further into factoring strategies in the following topic.

**Factored form** will always look like:

$$y=(xpm h)(xpm k)$$

The key thing to remember with factoring is that when you **add** (h+k), it must equal the **coefficient **to *x *(–2 in the equation above). When you **multiply **(htimes k), it must equal the **constant **(–3 in the equation above). Therefore, *h *and *k *must equal (–3) and (1). So our equation in factored form would be:

$$y=(x-3)(x+1)$$

If we drew a graph of (y=(x-3)(x+1)), it would be exactly the same as the graph of (y=2x^2-4x-6) pictured above. The difference in the two equations is that we can easily figure out the *x*-intercepts if the equation is in factored form. The *x*-intercepts are the *x *coordinates when the *y *value is 0. Looking at the factored equation above, we can see that *y *will be 0 if *x *= 3 or *x *= -1. That means that the graph crosses over the *x*-axis at those two points.

**Practice:**

Practice factoring the following equations:

(y=x^2+8x+15)

(y=x^2+x-12)

(y=x^2-5x+6)

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