How many intercepts does $y = x^{2} - 6 x + 9$ $y = x^2 − 6x + 9$ have?

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Answer 1 · 2015-08-12T23:51:10

One $x$ $x$ -intercept and one $y$ $y$ -intercept.

You can determine $x$ $x$ -intercepts by making the function equal to zero and $y$ $y$ -intercepts by evaluating the function for $x = 0$ $x=0$ .

When $y = 0$ $y=0$ , you have

$x^{2} - 6 x + 9 = 0$ $x^2 - 6x + 9 = 0$

In order to determine how many solutions this quadratic equation has, you can calculate the value of its discriminant, $Delta$ .

For a quadratic equation that takes the general form

$color(blue)(ax^2 + bx + c = 0)$

the discriminant is equal to

$color(blue)(Delta = b^2 - 4ac)$

In your case, you have $a=1$ , $b=-6$ , and $c=9$ , which means that the discriminant is equal to

$Delta = (-6)^2 - 4 * 1 * 9$

$Delta = 36 - 36 = color(green)(0)$

When the discriminant is equal to zero, your equation will only have one real solution (a repeated root) that takes the form

$x = (-b +- sqrt(Delta))/(2a) = (-b +- 0)/(2a) = -b/(2a)$

In your case, the root will be

$x = -((-6))/(2 * 1) = 6/2 = 3$

This means that the function has one $x$ -intercept, $x=3$ .

The $y$ -intercept will be

$y = (0)^2 - 6 * (0) + 9 = 9$

The function will thus intercept the $y$ -axis in the point $(0,9)$ and the $x$ -acis in the point $(3,0)$ .

graph{x^2 - 6x + 9 [-10, 10, -5, 5]}

How many intercepts does y = x^2 − 6x + 9y=x2−6x+9y = x^2 − 6x + 9 have?