Explore Westonci.ca, the leading Q&A site where experts provide accurate and helpful answers to all your questions. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform.
Sagot :
To find the solutions to the equation [tex]\(x^3 + 6x^2 - 40x = 192\)[/tex], we need to understand where the graph of the function [tex]\( y = x^3 + 6x^2 - 40x \)[/tex] intersects the horizontal line [tex]\( y = 192 \)[/tex].
Let's rephrase the problem into one equation by setting the two equations equal to each other:
[tex]\[ x^3 + 6x^2 - 40x = 192 \][/tex]
But for clarity of understanding, we'll convert it to:
[tex]\[ x^3 + 6x^2 - 40x - 192 = 0 \][/tex]
Now, let's solve this polynomial equation step-by-step to find the values of [tex]\( x \)[/tex].
### Step-by-Step Solution
1. Equation Setup:
[tex]\[ x^3 + 6x^2 - 40x - 192 = 0 \][/tex]
2. Finding Rational Roots:
By the Rational Root Theorem, potential rational roots of the polynomial are the factors of the constant term (-192) divided by the factors of the leading coefficient (1). Hence, the possible rational roots are:
[tex]\[ \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 12, \pm 16, \pm 24, \pm 32, \pm 48, \pm 64, \pm 96, \pm 192 \][/tex]
3. Testing Potential Roots:
Start by checking some of the potential roots using substitution.
- Testing [tex]\( x = -8 \)[/tex]:
[tex]\[ (-8)^3 + 6(-8)^2 - 40(-8) - 192 = -512 + 384 + 320 - 192 \][/tex]
[tex]\[ = 0 \][/tex]
Since this yields 0, [tex]\( x = -8 \)[/tex] is a root.
4. Polynomial Division:
Use polynomial long division or synthetic division to divide the original polynomial by [tex]\( (x + 8) \)[/tex], since [tex]\( x + 8 \)[/tex] is a factor.
Dividing [tex]\( x^3 + 6x^2 - 40x - 192 \)[/tex] by [tex]\( x + 8 \)[/tex] gives:
[tex]\[ x^3 + 6x^2 - 40x - 192 = (x + 8)(x^2 - 2x - 24) \][/tex]
5. Factoring the Quadratic Expression:
We now need to solve the quadratic equation [tex]\( x^2 - 2x - 24 = 0 \)[/tex].
To factor this quadratic, find two numbers that multiply to -24 and add to -2.
[tex]\[ x^2 - 2x - 24 = (x - 6)(x + 4) \][/tex]
6. Finding Additional Roots:
Set the factors equal to zero:
[tex]\[ x - 6 = 0 \quad \Rightarrow \quad x = 6 \][/tex]
[tex]\[ x + 4 = 0 \quad \Rightarrow \quad x = -4 \][/tex]
### Summary and Verification
We have found the roots of the polynomial: [tex]\(x = -8\)[/tex], [tex]\(x = -4\)[/tex], and [tex]\(x = 6\)[/tex]. Verify these solutions by substituting back into the original equation:
[tex]\[ x^3 + 6x^2 - 40x - 192 = 0 \][/tex]
- For [tex]\( x = -8 \)[/tex]:
[tex]\[ (-8)^3 + 6(-8)^2 - 40(-8) - 192 = -512 + 384 + 320 - 192 = 0 \][/tex]
- For [tex]\( x = -4 \)[/tex]:
[tex]\[ (-4)^3 + 6(-4)^2 - 40(-4) - 192 = -64 + 96 + 160 - 192 = 0 \][/tex]
- For [tex]\( x = 6 \)[/tex]:
[tex]\[ 6^3 + 6(6^2) - 40(6) - 192 = 216 + 216 - 240 - 192 = 0 \][/tex]
Thus, the solutions to the equation [tex]\( x^3 + 6x^2 - 40x = 192 \)[/tex] are:
[tex]\[ \boxed{x = -8, x = -4, x = 6} \][/tex]
Let's rephrase the problem into one equation by setting the two equations equal to each other:
[tex]\[ x^3 + 6x^2 - 40x = 192 \][/tex]
But for clarity of understanding, we'll convert it to:
[tex]\[ x^3 + 6x^2 - 40x - 192 = 0 \][/tex]
Now, let's solve this polynomial equation step-by-step to find the values of [tex]\( x \)[/tex].
### Step-by-Step Solution
1. Equation Setup:
[tex]\[ x^3 + 6x^2 - 40x - 192 = 0 \][/tex]
2. Finding Rational Roots:
By the Rational Root Theorem, potential rational roots of the polynomial are the factors of the constant term (-192) divided by the factors of the leading coefficient (1). Hence, the possible rational roots are:
[tex]\[ \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 12, \pm 16, \pm 24, \pm 32, \pm 48, \pm 64, \pm 96, \pm 192 \][/tex]
3. Testing Potential Roots:
Start by checking some of the potential roots using substitution.
- Testing [tex]\( x = -8 \)[/tex]:
[tex]\[ (-8)^3 + 6(-8)^2 - 40(-8) - 192 = -512 + 384 + 320 - 192 \][/tex]
[tex]\[ = 0 \][/tex]
Since this yields 0, [tex]\( x = -8 \)[/tex] is a root.
4. Polynomial Division:
Use polynomial long division or synthetic division to divide the original polynomial by [tex]\( (x + 8) \)[/tex], since [tex]\( x + 8 \)[/tex] is a factor.
Dividing [tex]\( x^3 + 6x^2 - 40x - 192 \)[/tex] by [tex]\( x + 8 \)[/tex] gives:
[tex]\[ x^3 + 6x^2 - 40x - 192 = (x + 8)(x^2 - 2x - 24) \][/tex]
5. Factoring the Quadratic Expression:
We now need to solve the quadratic equation [tex]\( x^2 - 2x - 24 = 0 \)[/tex].
To factor this quadratic, find two numbers that multiply to -24 and add to -2.
[tex]\[ x^2 - 2x - 24 = (x - 6)(x + 4) \][/tex]
6. Finding Additional Roots:
Set the factors equal to zero:
[tex]\[ x - 6 = 0 \quad \Rightarrow \quad x = 6 \][/tex]
[tex]\[ x + 4 = 0 \quad \Rightarrow \quad x = -4 \][/tex]
### Summary and Verification
We have found the roots of the polynomial: [tex]\(x = -8\)[/tex], [tex]\(x = -4\)[/tex], and [tex]\(x = 6\)[/tex]. Verify these solutions by substituting back into the original equation:
[tex]\[ x^3 + 6x^2 - 40x - 192 = 0 \][/tex]
- For [tex]\( x = -8 \)[/tex]:
[tex]\[ (-8)^3 + 6(-8)^2 - 40(-8) - 192 = -512 + 384 + 320 - 192 = 0 \][/tex]
- For [tex]\( x = -4 \)[/tex]:
[tex]\[ (-4)^3 + 6(-4)^2 - 40(-4) - 192 = -64 + 96 + 160 - 192 = 0 \][/tex]
- For [tex]\( x = 6 \)[/tex]:
[tex]\[ 6^3 + 6(6^2) - 40(6) - 192 = 216 + 216 - 240 - 192 = 0 \][/tex]
Thus, the solutions to the equation [tex]\( x^3 + 6x^2 - 40x = 192 \)[/tex] are:
[tex]\[ \boxed{x = -8, x = -4, x = 6} \][/tex]
We appreciate your time on our site. Don't hesitate to return whenever you have more questions or need further clarification. Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Westonci.ca is committed to providing accurate answers. Come back soon for more trustworthy information.
