Use any method to locate the zeros of y=−3x2+12x+15

Question

Use any method to locate the zeros of y=−3x2+12x+15
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GPT-4o mini · Answer

To locate the zeros of the quadratic function $y = - 3 x^{2} + 12 x + 15$ , we need to find the values of $x$ where $y = 0$ . This means we need to solve the equation:

$- 3 x^{2} + 12 x + 15 = 0$

We can use the quadratic formula, which is:

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

In this case, the coefficients $a$ , $b$ , and $c$ are:

$a = - 3$
$b = 12$
$c = 15$

Now, we will plug these values into the quadratic formula.

Calculate the discriminant $b^{2} - 4 a c$ :

$b^{2} = 12^{2} = 144$ $4 a c = 4 (- 3) (15) = - 180$ $b^{2} - 4 a c = 144 - (- 180) = 144 + 180 = 324$

Now plug the values into the quadratic formula:

$x = \frac{- 12 \pm \sqrt{324}}{2 (- 3)}$

Calculate $\sqrt{324}$ :

$\sqrt{324} = 18$

Substitute back into the formula:

$x = \frac{- 12 \pm 18}{- 6}$

Calculate the two possible values for $x$ :

First solution:

$x = \frac{- 12 + 18}{- 6} = \frac{6}{- 6} = - 1$

Second solution:

$x = \frac{- 12 - 18}{- 6} = \frac{- 30}{- 6} = 5$

Thus, the zeros of the function $y = - 3 x^{2} + 12 x + 15$ are:

$x = - 1 and x = 5$

Summary:

The zeros of the function are $x = - 1$ and $x = 5$ .