a(x)=−(x+6)^2+6. Find the x-intercepts, if any. Express the intercept(s) as ordered pair(s).

Question

Find the x-intercepts, if any. Express the intercept(s) as ordered pair(s).

Answer 1

To find the x-intercepts of the function a(x) = -(x + 6)^2 + 6, we set a(x) = 0 and solve for x. This is because the x-intercepts occur where the function equals zero.

1. Start with the equation:

     0 = -(x + 6)^2 + 6

2. Subtract 6 from both sides:

     -(x + 6)^2 = -6

3. Divide both sides by -1:

     (x + 6)^2 = 6

4. Take the square root of both sides:

     x + 6 = ±√6

5. Solve for x by subtracting 6 from both sides:

      x = -6 ± √6

Thus, the x-intercepts are:

x₁ = -6 + √6 and x₂ = -6 - √6

The x-intercepts as ordered pairs are:

(-6 + √6, 0) and (-6 - √6, 0)

Answer 2

The x-intercepts are located at points where y = 0:

0=-(x+6)² + 6

-(x+6)² = -6

(x+6)² = 6

x+6 = ±√6

x = -6 ± √6

So the x-intercepts are located at (-6+√6,0) and (-6-√6,0)

Check it out here:

desmos.com/calculator/43fbnlkqcl

Answer 3

Great question!

Quadratics can be in several different forms. Currently, the given quadratic is in vertex form. To identify the x-intercepts, it would be most helpful to transform it into intercept form.

To accomplish this, we'll first want to expand the binomial by squaring it:

(x + 6)(x + 6) = x² + 6x + 6x + 36 =

Putting that back in the equation, we'll then need to distribute the negative and combine like terms:

-(x² + 12x + 36) + 6 = -x² - 12x - 36 + 6 = -x² - 12x - 30

Now that it's in standard form, to put it in intercept form, we'll need to factor it. Unfortunately, if we try to factor it, we'll discover it is prime and therefore unfactorable. Instead we'll need to use the quadratic formula to find the x-intercepts:

[ -(-12)±√((-12)²-4(-1)(-30)) ] / 2(-1) = [12 ± √(144-120) ] /-2 = [ 12 ± √(24) ] / -2 = [12 ± 2√6 ] / -2

= -6 ± √6

This gives both x-intercepts: (-6 + √6 , 0) and (-6 - √6 , 0)

A bit trickier than if we could factor, but I hope this helps!