What are the solutions of the equation x4 – 5×2 – 36 = 0? use factoring to solve.
The Correct Answer and Explanation is :
The given equation is:
[
x^4 – 5x^2 – 36 = 0
]
To solve this equation by factoring, follow these steps:
Step 1: Make a substitution
Notice that the equation contains (x^4) and (x^2), which suggests that substituting a new variable might simplify the equation. Let:
[
y = x^2
]
This transforms the equation into:
[
y^2 – 5y – 36 = 0
]
Now, you have a quadratic equation in terms of (y).
Step 2: Factor the quadratic equation
We need to factor the quadratic equation (y^2 – 5y – 36 = 0). To factor, we look for two numbers that multiply to give (-36) (the constant term) and add to give (-5) (the coefficient of (y)).
The two numbers that satisfy this condition are (-9) and (4) because:
[
-9 \times 4 = -36 \quad \text{and} \quad -9 + 4 = -5
]
So, we can factor the quadratic as:
[
(y – 9)(y + 4) = 0
]
Step 3: Solve for (y)
Now that we have the factored form, set each factor equal to zero:
[
y – 9 = 0 \quad \text{or} \quad y + 4 = 0
]
Solving these equations gives:
[
y = 9 \quad \text{or} \quad y = -4
]
Step 4: Substitute back for (x)
Recall that (y = x^2), so we now substitute back:
- If (y = 9), then:
[
x^2 = 9
]
Taking the square root of both sides:
[
x = \pm 3
]
- If (y = -4), then:
[
x^2 = -4
]
This has no real solutions because the square of a real number cannot be negative. Thus, there are no real solutions from this equation.
Final Answer
The real solutions to the original equation are:
[
x = 3 \quad \text{or} \quad x = -3
]