What are the solutions of the equation x4 – 5×2 – 36 = 0? use factoring to solve.

1. $x=-3,\; 3$
2. $x=-4,9$
3. $x=4,-9$
4. $x=0$

The Correct Answer is

$a.\; x=-3,\; 3$

Correct Answer Explanation: $a.\; x=-3,\; 3$

Let’s solve the given quadratic equation x4−5x2−36=0 using factoring.

Step 1: Recognizing the Quadratic Form

Notice that this equation is in quadratic form, meaning we can represent it as (x2)2−5(x2)−36=0.

Step 2: Factoring

Now, let’s introduce a substitution to make this equation easier to factor. Let y=x2, so our equation becomes y2−5y−36=0.

Now, we need to factor the quadratic expression y2−5y−36. We are looking for two numbers whose product is $-36\times 1$ (coefficient of y2×constant term) and whose sum is $-5$ (coefficient of $y$).

The numbers that satisfy this condition are $-9$ and $4$ because $-9\times 4=-36$ and $-9+4=-5$.

So, we can express y2−5y−36 as $(y-9)(y+4)=0$.

Step 3: Substitute Back

Now, substitute back x2 for $y$: (x2−9)(x2+4)=0.

Now, we have a product of two factors equal to zero. According to the zero-product property, this implies that either x2−9=0 or x2+4=0.

Solving for x2−9=0:

x2−9=0

Add 9 to both sides:

x2=9

Take the square root of both sides:

$x=\pm 3$

Solving for x2+4=0:

x2+4=0

Subtract 4 from both sides:

x2=−4

This has no real solutions because the square of any real number is non-negative, and here we’re looking for a square to be $-4$.

The solutions to the original equation x4−5x2−36=0 are $x=-3,3$ based on the solutions obtained from x2−9=0.

Why the other solutions are  Incorrect:

b. Incorrect Solution: $x=-4,9$

This solution is incorrect because it does not satisfy the original equation x4−5x2−36=0. If we substitute $x=-4$ or $x=9$ into the original equation, we get $256-5(16)-36$ or $81-5(81)-36$, neither of which equals zero.

c.  Incorrect Solution: $x=4,-9$

Similarly, this solution is incorrect. Substituting $x=4$ or $x=-9$ into the original equation results in $256-5(16)-36$ or $81-5(81)-36$, which are not equal to zero.

d.  Incorrect Solution: $x=0$

This solution is also incorrect. When substituting $x=0$ into the original equation, we get $(-36)$, which is not equal to zero.

All the incorrect solutions were evaluated by substituting them into the original equation x4−5x2−36=0, and in each case, the equation did not hold true. Therefore, the correct solutions are $x=-3,3$, and the incorrect solutions are $x=-4,9$ and $x=4,-9$, and $x=0$.

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Smirti

(Founder of Management Notes)
MBA,BBA.

I am Smirti Bam, an enthusiastic edu blogger with a passion for sharing insights into the dynamic world of business and management through this website.

I hold a MBA degree from Presidential Business School, Kathmandu, and a BBA degree with a specialization in Finance from Apex College,

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