Question

Graph the function to find the zeros. Rewrite the function with the polynomial in factored form.yequals2 x squared plus 5 x minus 7Question content area bottomPart 1The zeros of the function areenter your response here.​(Use a comma to separate​ answers.)

Answer 1

To find the zeros of the function \( y = 2x^2 + 5x - 7 \), we can either graph the function or use the quadratic formula. Let's first find the zeros using the quadratic formula:

The quadratic formula is given by:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

For the function \( y = 2x^2 + 5x - 7 \):

- \( a = 2 \)
- \( b = 5 \)
- \( c = -7 \)

1. Calculate the discriminant \( b^2 - 4ac \):

\[
b^2 - 4ac = 5^2 - 4 \cdot 2 \cdot (-7) = 25 + 56 = 81
\]

2. Substitute the values into the quadratic formula:

\[
x = \frac{-5 \pm \sqrt{81}}{2 \cdot 2}
\]

\[
x = \frac{-5 \pm 9}{4}
\]

This gives us two solutions:

- For \( x = \frac{-5 + 9}{4} = \frac{4}{4} = 1 \)
- For \( x = \frac{-5 - 9}{4} = \frac{-14}{4} = -\frac{7}{2} \)

Thus, the zeros of the function are:

\[
1, -\frac{7}{2}
\]

---

Now, to rewrite the function in factored form, we can use the zeros we found. The factored form of a quadratic function is given as:

\[
y = a(x - r_1)(x - r_2)
\]

Where \( r_1 \) and \( r_2 \) are the roots (zeros) of the function. Here \( r_1 = 1 \) and \( r_2 = -\frac{7}{2} \). Therefore, we express \( y \) as:

\[
y = 2(x - 1)\left(x + \frac{7}{2}\right)
\]

To clear the fraction in \( x + \frac{7}{2} \), multiply by 2:

\[
y = 2(x - 1)\left(\frac{2x + 7}{2}\right) = (x - 1)(2x + 7)
\]

Thus, the final factored form is:

\[
y = 2(x - 1)\left(x + \frac{7}{2}\right)
\]
or simply

\[
y = 2\left(x - 1\right)\left(2x + 7\right)
\]

So the final answers are:
- The zeros of the function: \( \boxed{1, -\frac{7}{2}} \)
- The factored form: \( y = 2(x - 1)\left(2x + 7\right) \)