Question 104476This question is from textbook Beginning Algebra
: Solve the following quadratic equations.
4x^2=13x+12
-4x^2+13x+12=0
(-1+4)(4+3)=0
Honestly, I have no clue here. I tried to factor and solve by using the zero product principle, but really got lost. If you could, could you please provide me with details on how the solution is reached. I really want to learn this, and don't just want the answer.
This question is from textbook Beginning Algebra
Found 2 solutions by HyperBrain, stormy:
Answer by HyperBrain(694)   (Show Source):
Answer by stormy(4) (Show Source):
You can put this solution on YOUR website! You were heading in the right direction, but you didn't quite get there.
First, put the equation in the form of Ax^2 + Bx + C = 0 by moving all the terms to the left side of the equation:
4x^2 - 13x - 12 = 0
Note that A = 4, B = -13, and C = -12
Next, create your parentheses and place the variable in the proper location for solving using the FOIL method:
(Fx + O)(Ix + L) = 0
Now comes the time consuming part. You need to test all possible combinations of A and C until you find the combination that makes the following equations true:
A = F x I
C = O x L
B =(F x L) + (O x I)
Note:
F x I is also known as the First two terms
O x L is also known as the Last two terms
(F x L) + (O x I) is also known as Outer + Inner terms
Possible test combinations for our problem are:
A = 4: Factors into 1 x 4 and 2 x 2
C = -12: Factors into 1 x -12, -1 x 12, 2 x -6, -2 x 6, 3 x -4, and -3 x 4
Note that reverse combinations should also be tried, such as (4 x 1) for A.
So let's try out a combination (Remember A = 4, B = -13, and C = -12):
A = F x I = 1 x 4 = 4, which is correct
C = O x L = 1 x -12 = -12, which is correct
B = (F x L) + (O x I) = (1 x -12) + (1 x 4) = -8, which is incorrect
Try another combination:
A = F x I = 1 x 4 = 4, which is correct
C = O x L = -12 x 1 = -12, which is correct
B = (F x L) + (O x I) = (1 x 1) + (-12 x 4) = -47, which is incorrect
Hmmm... This could take some time. Try another combination:
A = F x I = 1 x 4 = 4, which is correct
C = O x L = -4 x 3 = -12, which is correct
B = (F x L) + (O x I) = (1 x 3) + (-4 x 4) = -13, which is correct!
Now plug the values for F, O, I and L back into the FOIL equation from above:
(Fx + O)(Ix + L) = 0
(1x + (-4))(4x + 3) = 0
(x - 4)(4x + 3) = 0
The only thing left to do is to solve for x (this is where the Zero Product Principle is used: If AB = 0, then A = 0 or B = 0. Or both could be 0.) To do this, set each expression that is inside a parenthesis equal to 0 and solve for x:
x - 4 = 0
x = 4
4x + 3 = 0
4x = -3
x = -3/4, or -0.75
To check your answers, plug the x values found back into the original equation and see if the equation is true:
4(4)^2 - 13(4) - 12 = 64 - 52 - 12 = 0 True!
4(-0.75)^2 - 13(-0.75) - 12 = 2.25 + 9.75 - 12 = 0 True!
Don't get discouraged if the answer doesn't pop out at you after a few tries when you are using the FOIL method. Just keep chugging. As long as you don't make any mistakes, you will eventually find the answer to this type of a problem. Happy hunting!
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