Question

interpreting the zeroes of a quadratic function
$y = -6x^2 + 100x - 180$
$x = $ selling price of each soccer ball
$y = $ daily profit from soccer balls
what are the zeroes of the function? round to the nearest hundredth.
$\square$ 0
$\square$ 2.05
$\square$ 8.33
$\square$ 14.61

Explanation:

Step1: Set \( y = 0 \) for zeroes

To find the zeroes of the quadratic function \( y=-6x^{2}+100x - 180 \), we set \( y = 0 \), so we have the equation:

$$ -6x^{2}+100x - 180=0 $$

We can simplify this equation by dividing all terms by -2:

$$ 3x^{2}-50x + 90=0 $$

Step2: Use quadratic formula

For a quadratic equation \( ax^{2}+bx + c = 0 \) (here \( a = 3 \), \( b=-50 \), \( c = 90 \)), the quadratic formula is \( x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a} \).

First, calculate the discriminant \( D=b^{2}-4ac \):

$$ D=(-50)^{2}-4\times3\times90=2500 - 1080 = 1420 $$

Then, find the roots:

$$ x=\frac{50\pm\sqrt{1420}}{2\times3}=\frac{50\pm\sqrt{4\times355}}{6}=\frac{50\pm2\sqrt{355}}{6}=\frac{25\pm\sqrt{355}}{3} $$

Calculate the numerical values:

• For the plus sign: \( x=\frac{25+\sqrt{355}}{3}\approx\frac{25 + 18.84}{3}=\frac{43.84}{3}\approx14.61 \)
• For the minus sign: \( x=\frac{25-\sqrt{355}}{3}\approx\frac{25 - 18.84}{3}=\frac{6.16}{3}\approx2.05 \)

Answer:

The zeroes of the function are approximately \( 2.05 \) and \( 14.61 \). So the correct options are:

• 2.05
• 14.61