Welcome to Westonci.ca, where your questions are met with accurate answers from a community of experts and enthusiasts. Join our platform to connect with experts ready to provide precise answers to your questions in various areas. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.
Sagot :
Let's analyze each equation to determine whether it is quadratic. Quadratic equations are in the form [tex]\( ax^2 + bx + c = 0 \)[/tex], where [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are constants, and [tex]\( a \neq 0 \)[/tex].
1. Equation: [tex]\(5x^2 + 15x = 0\)[/tex]
This equation is already in the standard form [tex]\(5x^2 + 15x = 0\)[/tex], which matches the quadratic form [tex]\( ax^2 + bx + c = 0 \)[/tex] with [tex]\( a = 5 \)[/tex], [tex]\( b = 15 \)[/tex], and [tex]\( c = 0 \)[/tex].
Therefore, [tex]\(5x^2 + 15x = 0\)[/tex] is a quadratic equation.
2. Equation: [tex]\(6x - 1 = 4x + 7\)[/tex]
Simplifying this equation:
[tex]\[ 6x - 4x - 1 = 7 \implies 2x - 1 = 7 \implies 2x = 8 \implies x = 4 \][/tex]
This is a linear equation because it does not contain an [tex]\( x^2 \)[/tex] term.
Therefore, [tex]\(6x - 1 = 4x + 7\)[/tex] is not a quadratic equation.
3. Equation: [tex]\(x^2 - 4x = 4x + 7\)[/tex]
Simplifying this equation:
[tex]\[ x^2 - 4x - 4x = 7 \implies x^2 - 8x = 7 \implies x^2 - 8x - 7 = 0 \][/tex]
This is in the standard form of a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] with [tex]\( a = 1 \)[/tex], [tex]\( b = -8 \)[/tex], and [tex]\( c = -7 \)[/tex].
Therefore, [tex]\(x^2 - 4x = 4x + 7\)[/tex] is a quadratic equation.
4. Equation: [tex]\(2x - 1 = 0\)[/tex]
This equation is in the form [tex]\(2x - 1 = 0\)[/tex], which is a linear equation with no [tex]\( x^2 \)[/tex] term.
Therefore, [tex]\(2x - 1 = 0\)[/tex] is not a quadratic equation.
5. Equation: [tex]\(3x^2 + 5x - 7 = 0\)[/tex]
This equation is already in the standard form [tex]\(3x^2 + 5x - 7 = 0\)[/tex], which matches the quadratic form [tex]\( ax^2 + bx + c = 0 \)[/tex] with [tex]\( a = 3 \)[/tex], [tex]\( b = 5 \)[/tex], and [tex]\( c = -7 \)[/tex].
Therefore, [tex]\(3x^2 + 5x - 7 = 0\)[/tex] is a quadratic equation.
6. Equation: [tex]\(x^3 - 2x^2 + 1 = 0\)[/tex]
This equation is in the form [tex]\(x^3 - 2x^2 + 1 = 0\)[/tex], which contains a cubic term [tex]\( x^3 \)[/tex]. Since it has a term with [tex]\( x \)[/tex] raised to the third power, it is not a quadratic equation.
Therefore, [tex]\(x^3 - 2x^2 + 1 = 0\)[/tex] is not a quadratic equation.
In summary, the following equations are quadratic:
- [tex]\(5x^2 + 15x = 0\)[/tex]
- [tex]\(x^2 - 4x = 4x + 7\)[/tex]
- [tex]\(3x^2 + 5x - 7 = 0\)[/tex]
So, the quadratic equations are:
1. [tex]\(5 x^2 + 15 x = 0\)[/tex]
2. [tex]\(x^2 - 4 x = 4 x + 7\)[/tex]
3. [tex]\(3 x^2 + 5 x - 7 = 0\)[/tex]
1. Equation: [tex]\(5x^2 + 15x = 0\)[/tex]
This equation is already in the standard form [tex]\(5x^2 + 15x = 0\)[/tex], which matches the quadratic form [tex]\( ax^2 + bx + c = 0 \)[/tex] with [tex]\( a = 5 \)[/tex], [tex]\( b = 15 \)[/tex], and [tex]\( c = 0 \)[/tex].
Therefore, [tex]\(5x^2 + 15x = 0\)[/tex] is a quadratic equation.
2. Equation: [tex]\(6x - 1 = 4x + 7\)[/tex]
Simplifying this equation:
[tex]\[ 6x - 4x - 1 = 7 \implies 2x - 1 = 7 \implies 2x = 8 \implies x = 4 \][/tex]
This is a linear equation because it does not contain an [tex]\( x^2 \)[/tex] term.
Therefore, [tex]\(6x - 1 = 4x + 7\)[/tex] is not a quadratic equation.
3. Equation: [tex]\(x^2 - 4x = 4x + 7\)[/tex]
Simplifying this equation:
[tex]\[ x^2 - 4x - 4x = 7 \implies x^2 - 8x = 7 \implies x^2 - 8x - 7 = 0 \][/tex]
This is in the standard form of a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] with [tex]\( a = 1 \)[/tex], [tex]\( b = -8 \)[/tex], and [tex]\( c = -7 \)[/tex].
Therefore, [tex]\(x^2 - 4x = 4x + 7\)[/tex] is a quadratic equation.
4. Equation: [tex]\(2x - 1 = 0\)[/tex]
This equation is in the form [tex]\(2x - 1 = 0\)[/tex], which is a linear equation with no [tex]\( x^2 \)[/tex] term.
Therefore, [tex]\(2x - 1 = 0\)[/tex] is not a quadratic equation.
5. Equation: [tex]\(3x^2 + 5x - 7 = 0\)[/tex]
This equation is already in the standard form [tex]\(3x^2 + 5x - 7 = 0\)[/tex], which matches the quadratic form [tex]\( ax^2 + bx + c = 0 \)[/tex] with [tex]\( a = 3 \)[/tex], [tex]\( b = 5 \)[/tex], and [tex]\( c = -7 \)[/tex].
Therefore, [tex]\(3x^2 + 5x - 7 = 0\)[/tex] is a quadratic equation.
6. Equation: [tex]\(x^3 - 2x^2 + 1 = 0\)[/tex]
This equation is in the form [tex]\(x^3 - 2x^2 + 1 = 0\)[/tex], which contains a cubic term [tex]\( x^3 \)[/tex]. Since it has a term with [tex]\( x \)[/tex] raised to the third power, it is not a quadratic equation.
Therefore, [tex]\(x^3 - 2x^2 + 1 = 0\)[/tex] is not a quadratic equation.
In summary, the following equations are quadratic:
- [tex]\(5x^2 + 15x = 0\)[/tex]
- [tex]\(x^2 - 4x = 4x + 7\)[/tex]
- [tex]\(3x^2 + 5x - 7 = 0\)[/tex]
So, the quadratic equations are:
1. [tex]\(5 x^2 + 15 x = 0\)[/tex]
2. [tex]\(x^2 - 4 x = 4 x + 7\)[/tex]
3. [tex]\(3 x^2 + 5 x - 7 = 0\)[/tex]
Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.
