Question

How many times does the graph of 4x = 32 - x2 cross the x-axis?

0
1
2
What are the solutions to the equation?

Answer 1

2

Explanation:

4x = 32 - x2

x^2 + 4x - 32 = 0

x^2 + 8x - 4x - 32 = 0

x(x + 8) - 4(x + 8) = 0

(x + 8)(x - 4) = 0

so x = -8 and x = 4

So the graph crosses the x-axis twice at -8 and 4.

Answer 2

The graph crosses the x-axis 2 times

The solutions are x = -8 & x = 4

Explanation:

Qaudratics are in the form
`ax^2 + bx+ c`

Where a, b, c are constants

Now, let's arrange this equation in this form:

`4x=32-x^2\\x^2+4x-32=0`

Where

a = 1

b = 4

c = -32

We need to know the discriminant to know nature of roots. The discriminant is:

`D=b^2-4ac`

If

• D = 0 , we have 2 similar root and there is 2 solutions and that touches the x-axis
• D > 0, we have 2 distinct roots/solutions and both cut the x-axis
• D < 0, we have imaginary roots and it never cuts the x-axis

Let's find value of Discriminant:

`D=b^2-4ac\\D=(4)^2 -4(1)(-32)\\D=144`

Certainly D > 0, so there are 2 distinct roots and cuts the x-axis twice.

We get the roots/solutions by factoring:

`x^2+4x-32=0\\(x+8)(x-4)=0\\x=4,-8`

Thus,

The graph crosses the x-axis 2 times

The solutions are x = -8 & x = 4