How do you solve and check the solutions of:
x^2 + x - 12 / x - 3 = 15
and
2x^2 + 8x - 10 / 2x^2 + 14x + 20 = 4
I will do the hard one, you do the easier one
2x^2 + 8x - 10 / 2x^2 + 14x + 20 = 4
You probably meant:
(2x^2 + 8x - 10) / (2x^2 + 14x + 20) = 4 , those brackets are essential
2(x^2 + 4x - 5)/(2(x^2 + 7x + 10) = 4
(x+5)(x-1)/((x+5)(x+2)) = 4
(x-1)/(x+2) = 4 , as long as x ≠ -5
4x + 8 = x - 1
3x = -9
x = -3
LOL! Thanks Reiny! I totally missed where those brackets would be : )
To solve and check the solutions of these rational expressions, we first need to bring them to a common denominator and then solve for x. Let's start with the first equation:
x^2 + x - 12 / x - 3 = 15
To simplify the expression, we factor the numerator:
(x - 3)(x + 4) / x - 3 = 15
Next, we multiply both sides of the equation by (x - 3) to clear the denominator:
(x - 3)(x + 4) = 15(x - 3)
Expanding both sides gives:
x^2 + x - 12 = 15x - 45
Moving all terms to one side:
x^2 + x - 15x - 12 + 45 = 0
x^2 - 14x + 33 = 0
Now we have a quadratic equation, which we can solve by factoring or using the quadratic formula. In this case, the equation factors:
(x - 11)(x - 3) = 0
Setting each factor equal to zero gives us two possible solutions:
x - 11 = 0 --> x = 11
x - 3 = 0 --> x = 3
Now, let's check the solutions by substituting them back into the original equation one at a time:
For x = 11:
(11^2 + 11 - 12) / (11 - 3) = 15
(121 + 11 - 12) / 8 = 15
120 / 8 = 15
15 = 15 (True)
For x = 3:
(3^2 + 3 - 12) / (3 - 3) = 15
(9 + 3 - 12) / 0 = 15
0 / 0 = 15
Since 0 divided by 0 is undefined, x = 3 is not a valid solution.
Therefore, the only solution to the first equation is x = 11.
Now, let's solve the second equation:
2x^2 + 8x - 10 / 2x^2 + 14x + 20 = 4
Simplifying the expression:
(x^2 + 4x - 5) / (x^2 + 7x + 10) = 2
Factoring the numerator and denominator:
((x - 1)(x + 5)) / ((x + 2)(x + 5)) = 2
Canceling out common factors, we have:
(x - 1) / (x + 2) = 2
Next, we cross-multiply:
(x - 1) * 1 = (x + 2) * 2
Expanding both sides gives:
x - 1 = 2x + 4
Bringing all terms to one side:
x - 2x = 4 + 1
-x = 5
Dividing by -1 gives:
x = -5
Now, let's check the solution by substituting it back into the original equation:
For x = -5:
(2(-5)^2 + 8(-5) - 10) / (2(-5)^2 + 14(-5) + 20) = 4
(2(25) - 40 - 10) / (2(25) - 70 + 20) = 4
(50 - 40 - 10) / (50 - 70 + 20) = 4
0 / 0 = 4
Again, we have 0 divided by 0, which is undefined. Therefore, x = -5 is not a valid solution.
In conclusion, the first equation has one solution x = 11, and the second equation has no valid solutions.