Determine the zero(s) of the quadratic function y=6x2+13x+5 . Show all steps.

Answer 1

To determine the zeros of the quadratic function $y=6x^2+13x+5$, we can use the quadratic formula:

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

where $a$, $b$, and $c$ are the coefficients from the quadratic equation $a{x}^2+bx+c$. For the given function, we have:

• $a=6$
• $b=13$
• $c=5$

Step 1: Calculate the Discriminant

First, we will calculate the discriminant $D=b^2-4ac$:

$$D={13}^2-4\cdot 6\cdot 5$$ $$D=169-120$$ $$D=49$$

Step 2: Apply the Quadratic Formula

Now that we have $D=49$, we can substitute $a$, $b$, and $D$ into the quadratic formula:

$$x=\frac{-13\pm \sqrt{49}}{2\cdot 6}$$

Calculating the square root of 49:

$$\sqrt{49}=7$$

Now substituting back into the formula:

$$x=\frac{-13\pm 7}{12}$$

Step 3: Solve for the Two Solutions

Now we will calculate the two possible values for $x$:

1. First Solution $x_1$:

$$x_1=\frac{-13+7}{12}=\frac{-6}{12}=-\frac{1}{2}$$

2. Second Solution $x_2$:

$$x_2=\frac{-13-7}{12}=\frac{-20}{12}=-\frac{5}{3}$$

Conclusion

The zeros of the quadratic function $y=6x^2+13x+5$ are:

$$x=-\frac{1}{2}\text{and}x=-\frac{5}{3}$$