To determine the type of the equation [tex]\( y = 4x^2 - 5 \)[/tex], we need to analyze its structure and characteristics.
1. General Forms of Common Equations:
- Linear Equation: A linear equation in two variables (for example, [tex]\(y\)[/tex] and [tex]\(x\)[/tex]) has the form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] and [tex]\(b\)[/tex] are constants. The highest power of [tex]\(x\)[/tex] in a linear equation is 1.
- Quadratic Equation: A quadratic equation in one variable (like [tex]\(x\)[/tex]) has the form [tex]\(ax^2 + bx + c\)[/tex], where [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are constants and [tex]\(a \neq 0\)[/tex]. The highest power of [tex]\(x\)[/tex] in a quadratic equation is 2.
- Exponential Equation: An exponential equation has the form [tex]\(y = a \cdot b^x\)[/tex], where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are constants and [tex]\(b \neq 1\)[/tex]. The variable [tex]\(x\)[/tex] appears as an exponent.
2. Identifying the Given Equation:
The given equation is [tex]\(y = 4x^2 - 5\)[/tex].
3. Analyzing the Structure:
- Let’s identify the highest power of [tex]\(x\)[/tex] in the equation:
- In [tex]\( y = 4x^2 \)[/tex], the term [tex]\(4x^2\)[/tex] has the variable [tex]\(x\)[/tex] raised to the power of 2.
- The equation can be written in the standard form of a quadratic equation [tex]\(ax^2 + bx + c\)[/tex]:
- [tex]\(a = 4\)[/tex], [tex]\(b = 0\)[/tex], and [tex]\(c = -5\)[/tex].
- This matches the form [tex]\( ax^2 + bx + c \)[/tex] where [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are constants and the highest power of [tex]\(x\)[/tex] is 2.
4. Conclusion:
Since the highest power of [tex]\(x\)[/tex] in the equation [tex]\(y = 4x^2 - 5\)[/tex] is 2, and it fits the form of a quadratic equation, we can conclude that the given equation is quadratic.
Therefore, [tex]\( y = 4x^2 - 5 \)[/tex] is indeed a quadratic equation.
