Welcome to Westonci.ca, the place where your questions are answered by a community of knowledgeable contributors. Get detailed and precise answers to your questions from a dedicated community of experts on our Q&A platform. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.
Sagot :
To find the x-intercepts of the function \( f(x) = 4x^4 + 12x^3 - 40x^2 \), we need to solve for \( x \) when \( f(x) = 0 \).
1. Set the function equal to zero:
[tex]\[ 4x^4 + 12x^3 - 40x^2 = 0 \][/tex]
2. Factor out the greatest common factor (GCF):
Notice that \( 4x^2 \) is a common factor in each term of the polynomial. Factor out \( 4x^2 \):
[tex]\[ 4x^2(x^2 + 3x - 10) = 0 \][/tex]
3. Solve for \( x \) in \( 4x^2 = 0 \):
[tex]\[ 4x^2 = 0 \implies x^2 = 0 \implies x = 0 \][/tex]
Thus, one x-intercept is \( (0,0) \).
4. Solve the quadratic equation \( x^2 + 3x - 10 = 0 \):
Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = 3 \), and \( c = -10 \):
[tex]\[ x = \frac{-3 \pm \sqrt{3^2 - 4 \cdot 1 \cdot (-10)}}{2 \cdot 1} \][/tex]
[tex]\[ x = \frac{-3 \pm \sqrt{9 + 40}}{2} \][/tex]
[tex]\[ x = \frac{-3 \pm \sqrt{49}}{2} \][/tex]
[tex]\[ x = \frac{-3 \pm 7}{2} \][/tex]
This results in two solutions:
[tex]\[ x = \frac{-3 + 7}{2} = \frac{4}{2} = 2 \][/tex]
[tex]\[ x = \frac{-3 - 7}{2} = \frac{-10}{2} = -5 \][/tex]
Thus, the other two x-intercepts are \( (2,0) \) and \((-5,0) \).
5. Summarize the x-intercepts:
The x-intercepts of the function \( f(x) = 4x^4 + 12x^3 - 40x^2 \) are:
[tex]\[ (0,0), \, (2,0), \, (-5,0) \][/tex]
Therefore, the correct choice is:
[tex]\((0,0), (2,0), (-5,0)\)[/tex].
1. Set the function equal to zero:
[tex]\[ 4x^4 + 12x^3 - 40x^2 = 0 \][/tex]
2. Factor out the greatest common factor (GCF):
Notice that \( 4x^2 \) is a common factor in each term of the polynomial. Factor out \( 4x^2 \):
[tex]\[ 4x^2(x^2 + 3x - 10) = 0 \][/tex]
3. Solve for \( x \) in \( 4x^2 = 0 \):
[tex]\[ 4x^2 = 0 \implies x^2 = 0 \implies x = 0 \][/tex]
Thus, one x-intercept is \( (0,0) \).
4. Solve the quadratic equation \( x^2 + 3x - 10 = 0 \):
Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = 3 \), and \( c = -10 \):
[tex]\[ x = \frac{-3 \pm \sqrt{3^2 - 4 \cdot 1 \cdot (-10)}}{2 \cdot 1} \][/tex]
[tex]\[ x = \frac{-3 \pm \sqrt{9 + 40}}{2} \][/tex]
[tex]\[ x = \frac{-3 \pm \sqrt{49}}{2} \][/tex]
[tex]\[ x = \frac{-3 \pm 7}{2} \][/tex]
This results in two solutions:
[tex]\[ x = \frac{-3 + 7}{2} = \frac{4}{2} = 2 \][/tex]
[tex]\[ x = \frac{-3 - 7}{2} = \frac{-10}{2} = -5 \][/tex]
Thus, the other two x-intercepts are \( (2,0) \) and \((-5,0) \).
5. Summarize the x-intercepts:
The x-intercepts of the function \( f(x) = 4x^4 + 12x^3 - 40x^2 \) are:
[tex]\[ (0,0), \, (2,0), \, (-5,0) \][/tex]
Therefore, the correct choice is:
[tex]\((0,0), (2,0), (-5,0)\)[/tex].
We hope this information was helpful. Feel free to return anytime for more answers to your questions and concerns. Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Thank you for trusting Westonci.ca. Don't forget to revisit us for more accurate and insightful answers.