Question

How do you solve `x^2 - 5x = 36`?

Answer 1

`x=-4,9`

Explanation:

Subtract 36 from both sides

`x^2-5x-36=0`

This is the equation of a quadratic so wee need the x-intercepts.

Two numbers that multiply to 36 are 4 and -9, and they add to -5 so your equation can be

`(x+4)(x-9)=0`

so `x=-4,9`

Answer 2

`x^2 - 5x = 36`
minus 36 (both sides)
`x^2 - 5x - 36 = 0`
Now, we need to find two numbers (for exampe a and b).
`a*b = -36` and `a+b = -5`
`a = -9`
`b = +4`
So, the solution is: `(x-9)*(x+4)=0`
We want these brackets to be equal to zero.
`x=9`
`x = -4`
Two solutions.

Answer 3

Solve by either factoring or using The Quadratic Formula to find `x={-4,9}`

Explanation:

The first step is to set the stated expression equal to zero. we will do this by subtracting 36 from both sides:

`x^2-5xcolor(red)(-36)=cancel(36color(red)(-36))`

`x^2-5x-36=0`

Now, we can either factor this equation OR use The Quadratic Formula to find the values of `x` that satisfy the equation. Let's start with factoring.

Based on the order of the equation (second order due to the presence of `x^2`), we will assume this will have two factors. Additionally, since the third factor is negative, we know that one factor is positive and the other is negative:

`(x+a)(x+b)=x^2+(a+b)x+ab`

Now, we just need to figure out what two numbers satisfy the following system:

`a+b=-5`
`ab=-36`

We know that 36 is a square, so a possible combination is 6 and -6. However, that would not satisfy the first equation. Other factors of -36 include:

`{+-36, +-1}`
`{+-18, +-2}`
`{+-12, +-3}`
`{+-9, +-4}`

The only set there that would satisfy the first equation though, is `{-9,4}`. Now that we have our factors, we can write the factored equation and evaluate:

`x^2-5x-36=(x+4)(x-9)=0`

When `x=-4`:

`(color(purple)(-4)+4)(color(purple)(-4)-9)=0`

`(0)(-13)=0 rArr 0=0`

When `x=9`:

`(color(purple)(9)+4)(color(purple)(9)-9)=0`

`(13)(0)=0 rArr 0=0`

We have our two solutions for `x`:

`color(green)(x={-4,9})`

Using The Quadratic Formula:

We plug in our equations coefficients into the formula:

`x=(-b+-sqrt(b^2-4ac))/(2a)`

`color(red)(a=1)`
`color(blue)(b=-5)`
`color(purple)(c=-36)`

`x=(-color(blue)((-5))+-sqrt(color(blue)((-5))^2-4color(red)((1))color(purple)((-36))))/(2color(red)((1)))`

`x=(5+-sqrt(25-4(-36)))/(2)`

`x=(5+-sqrt(25+144))/(2)`

`x=(5+-sqrt(169))/(2)`

`x=(5+-13)/(2)`

`x={(5-13)/(2),(5+13)/(2)}`

`x={-8/2,18/2}`

`color(green)(x={-4,9})`