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Magazine Chapter 8: Problem 76
The following exercises are not grouped by type. Solve each equation. $$\left(x^{2}+x\right)^{2}+12=8\left(x^{2}+x\right)$$
Short Answer
Expert verified
The solutions to the equation are \(x = 2, -3, 1, -2\).
Step by step solution
01
- Introduce a substitution
Let鈥檚 set a new variable to simplify the equation. Assign: \( u = x^2 + x \). This substitution transforms the equation into a simpler form.
02
- Rewrite the equation
Using the substitution \( u = x^2 + x \), the original equation \( \left(x^{2}+x\right)^{2}+12=8\left(x^{2}+x\right) \) becomes \( u^2 + 12 = 8u \).
03
- Rearrange into a standard quadratic form
Rewrite the equation in the standard quadratic form: \( u^2 - 8u + 12 = 0 \).
04
- Solve the quadratic equation
Solve the quadratic equation \( u^2 - 8u + 12 = 0 \) using the quadratic formula: \( u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Here, \( a = 1 \), \( b = -8 \), and \( c = 12 \).
05
- Apply the quadratic formula
Calculate the discriminant: \( (-8)^2 - 4 \cdot 1 \cdot 12 = 64 - 48 = 16 \). Then, find the solutions for \( u \): \( u = \frac{8 \pm \sqrt{16}}{2} \) which simplifies to \( u = \frac{8 \pm 4}{2} \). This yields two solutions: \( u = 6 \) and \( u = 2 \).
06
- Substitute back to find x
Substitute back \( u = x^2 + x \) into each solution. For \( u = 6 \): \( x^2 + x = 6 \), solve the quadratic equation: \( x^2 + x - 6 = 0 \). For \( u = 2 \): \( x^2 + x = 2 \), solve the quadratic equation: \( x^2 + x - 2 = 0 \).
07
- Solve quadratic equations for x
Use the quadratic formula for \( x^2 + x - 6 = 0 \): Discriminant \( = 1 + 24 = 25 \), Solutions: \( x = \frac{-1 \pm 5}{2} \) resulting in \( x = 2 \) or \( x = -3 \). For \( x^2 + x - 2 = 0 \): Discriminant \( = 1 + 8 = 9 \), Solutions: \{ x = \frac{-1 \pm 3}{2} \} resulting in \( x = 1 \) or \( x = -2 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Substitution Method
The substitution method simplifies complex equations by introducing a new variable. Start by setting a variable like this: let鈥檚 assign \( u = x^2 + x \). This reduces the given equation to a more manageable form.
For example, if we have \( (x^2 + x)^2 + 12 = 8(x^2 + x) \), it simplifies to \( u^2 + 12 = 8u \). This is because we replace \( x^2 + x \) with \( u \).
Using substitution, a polynomial equation transforms into a standard quadratic form, making it easier to solve. After solving for the new variable, substitute back to find the original variable.
For example, if we have \( (x^2 + x)^2 + 12 = 8(x^2 + x) \), it simplifies to \( u^2 + 12 = 8u \). This is because we replace \( x^2 + x \) with \( u \).
Using substitution, a polynomial equation transforms into a standard quadratic form, making it easier to solve. After solving for the new variable, substitute back to find the original variable.
Discriminant
The discriminant in a quadratic equation \( ax^2 + bx + c = 0 \) is found using the formula \( b^2 - 4ac \). This value helps determine the nature of the solutions:
- If the discriminant is positive, there are two distinct real solutions.
- If it is zero, there is exactly one real solution.
- If it is negative, there are no real solutions, only complex ones.
Quadratic Formula
The quadratic formula \( u = \frac{-b \, \text{卤} \, \text{鈭殅 (b^2 - 4ac)}{2a} \) can solve any quadratic equation \( ax^2 + bx + c = 0 \). Here鈥檚 how to apply it:
First, identify the coefficients: \( a = 1 \), \( b = -8 \), and \( c = 12 \). Next, calculate the discriminant which is already computed as 16.
Plugging these into the formula, we get:
\[ u = \frac{-(-8) \, \text{卤} \, \text{鈭殅 (16)}{2 \times 1} = \frac{8 \, \text{卤} \, 4}{2} = \frac{8 + 4}{2}, \frac{8 - 4}{2} = 6 \text{ or } 2 \] Thus, the solutions are \( u = 6 \) and \( u = 2 \).
First, identify the coefficients: \( a = 1 \), \( b = -8 \), and \( c = 12 \). Next, calculate the discriminant which is already computed as 16.
Plugging these into the formula, we get:
\[ u = \frac{-(-8) \, \text{卤} \, \text{鈭殅 (16)}{2 \times 1} = \frac{8 \, \text{卤} \, 4}{2} = \frac{8 + 4}{2}, \frac{8 - 4}{2} = 6 \text{ or } 2 \] Thus, the solutions are \( u = 6 \) and \( u = 2 \).
Solving Polynomial Equations
To solve polynomial equations like \( u^2 - 8u + 12 = 0 \), follow these steps:
- Rearrange the equation to the standard form \( ax^2 + bx + c = 0 \).
- Use the quadratic formula to find \( u \).
- Substitute back the original variable \( x \) if you used substitution.
- For \( u = 6 \): The equation \( x^2 + x - 6 = 0 \) has solutions \( x = 2 \) or \( x = -3 \).
- For \( u = 2 \): The equation \( x^2 + x - 2 = 0 \) has solutions \( x = 1 \) or \( x = -2 \).
