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

3 Answers
May 1, 2018

Answer:

#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#

May 1, 2018

#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.

May 1, 2018

Answer:

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})#