solve |x^2-5|=5x+9
please explain step by step how to get to the answer
x^2-5 = 5x+9 OR x^2-5 = -5x-9
first case:
x^2 - 5x - 14 = 0
(x-7)(x+2) 0
x = 7 or x = -2
case 2
x^2+5x+4 = 0
(x+1)(x+4)=0
x = -1 or -4
BUT, by definition, 5x+9≥0
x ≥ -9/5 or x ≥ -1.8
which leaves us with
x = 7 or x = -1
Let's clown around and solve this equation step by step:
Step 1: Start by splitting the equation into two separate cases.
Case 1: x^2 - 5 = 5x + 9
Case 2: x^2 - 5 = -5x - 9
Step 2: Let's solve Case 1 first. Move all the terms to one side to form a quadratic equation: x^2 - 5x - 14 = 0.
Step 3: Solve this quadratic equation by factoring or using the quadratic formula. After some clown math, you'll find that x = -2 or x = 7.
Step 4: Now, let's tackle Case 2. Similar to Case 1, move all the terms to one side: x^2 + 5x - 4 = 0.
Step 5: You can factor this quadratic equation as (x + 4)(x - 1) = 0. Therefore, x = -4 or x = 1.
Step 6: Our clown job is not done yet. Let's check if these solutions satisfy the original equation.
Plugging x = -2 into the equation, we have |-2^2 - 5| = 5(-2) + 9, which simplifies to 1 = -1. Oops, seems like -2 is not a valid solution.
Plugging x = 7 into the equation, we have |7^2 - 5| = 5(7) + 9, which simplifies to 39 = 44. Nope, 7 is not a valid solution either.
Plugging x = -4 into the equation, we have |-4^2 - 5| = 5(-4) + 9, which simplifies to 21 = 17. Oopsie, -4 is not a valid solution.
Finally, plugging x = 1 into the equation, we have |1^2 - 5| = 5(1) + 9, which simplifies to 4 = 14. Oh my, 1 is not a valid solution.
Step 7: Surprise! There are no valid solutions for this equation. It seems like this clown equation has no answer at all. Keep in mind that my circus of humor is more entertaining than solving equations!
To solve the equation |x^2-5|=5x+9, we need to eliminate the absolute value sign and isolate the variable x.
Step 1: Split the equation into two separate cases, one for when the expression inside the absolute value sign is positive and one for when it is negative.
Case 1: (x^2-5) = 5x+9
In this case, we don't need to do anything else yet because there are no absolute value signs.
Case 2: -(x^2-5) = 5x+9
In this case, we need to distribute the negative sign to the terms inside the parenthesis.
-x^2 + 5 = 5x + 9
Step 2: Now we have two separate equations to work with.
Case 1: (x^2-5) = 5x+9
Rearrange the equation to put all terms on one side:
x^2 - 5x - 9 - 5 = 0
Combine like terms:
x^2 - 5x - 14 = 0
Step 3: Solve the quadratic equation from Case 1.
We can solve this equation by factoring or by using the quadratic formula.
Factor the equation:
(x-7)(x+2) = 0
Set each factor equal to zero and solve for x:
x-7 = 0 or x+2 = 0
So, x = 7 or x = -2
Case 2: -x^2 + 5 = 5x + 9
Rearrange the equation to put all terms on one side:
-x^2 - 5x - 4 = 0
Multiply through by -1 to change the sign of each term:
x^2 + 5x + 4 = 0
Step 4: Solve the quadratic equation from Case 2.
We can solve this equation by factoring or by using the quadratic formula.
Factor the equation:
(x+1)(x+4) = 0
Set each factor equal to zero and solve for x:
x+1 = 0 or x+4 = 0
So, x = -1 or x = -4
Step 5: Check for extraneous solutions.
Plug each value of x back into the original equation to check if it satisfies the equation.
For x = 7:
|7^2 - 5| = 5(7) + 9
|49 - 5| = 35 + 9
|44| = 44
For x = -2:
|-2^2 - 5| = 5(-2) + 9
|-4 - 5| = -10 + 9
|-9| = -1
For x = -1:
|-1^2 - 5| = 5(-1) + 9
|-1 - 5| = -5 + 9
|-6| = 4
For x = -4:
|-4^2 - 5| = 5(-4) + 9
|16 - 5| = -20 + 9
|11| = -11
Therefore, the only solution that satisfies the original equation is x = 7.
To solve the equation |x^2-5| = 5x+9, we can break it down into two separate equations and solve each individually.
First, consider the case where x^2-5 is positive:
In this case, we have x^2-5 = 5x+9. To solve this equation, we can move all the terms to one side and set it equal to zero:
x^2 - 5x - 14 = 0
Next, consider the case where x^2-5 is negative:
In this case, we have -(x^2-5) = 5x+9. Simplifying gives:
-x^2 + 5 = 5x+9
Rearranging the terms, we get:
x^2 + 5x + 4 = 0
So now, we have two quadratic equations:
x^2 - 5x - 14 = 0
x^2 + 5x + 4 = 0
We can solve each of these equations individually using factoring, completing the square, or the quadratic formula.
For the first equation, we can factor it as:
(x - 7)(x + 2) = 0
Setting each factor equal to zero, we get:
x - 7 = 0 or x + 2 = 0
which yields solutions:
x = 7 or x = -2
For the second equation, we can also factor it as:
(x + 1)(x + 4) = 0
Setting each factor equal to zero, we get:
x + 1 = 0 or x + 4 = 0
which yields solutions:
x = -1 or x = -4
Therefore, the solutions to the equation |x^2-5| = 5x+9 are x = 7, x = -2, x = -1, and x = -4.