- Rearrange the equation into standard form: \(4x^2 - 32x + 64 = 0\).
- Identify coefficients: \(a = 4, b = -32, c = 64\).
- Calculate the discriminant: \(D = (-32)^2 - 4(4)(64) = 0\).
- Since \(D=0\), the equation has one real, rational root.
- Final Answer: \(\boxed{c}\)
Explanation
- Rearranging the Equation
To determine the nature of the solutions for the equation \(32x - 4 = 4x^2 + 60\), we first rewrite it in the standard quadratic form, \(ax^2 + bx + c = 0\). By moving all terms to one side, we get:
\[4x^2 - 32x + 64 = 0\]
Here, we can identify the coefficients as \(a = 4\), \(b = -32\), and \(c = 64\).
- Calculating the Discriminant
The discriminant, \(D\), of a quadratic equation \(ax^2 + bx + c = 0\) is defined by the formula:
\[D = b^2 - 4ac\]
Substituting our values \(a = 4\), \(b = -32\), and \(c = 64\):
\[D = (-32)^2 - 4 \times 4 \times 64\]
\[D = 1024 - 1024\]
\[D = 0\]
- Analyzing the Discriminant and Conclusion
The value of the discriminant determines the nature of the roots of a quadratic equation. If \(D > 0\), there are two real roots. If \(D < 0\), there are two complex (non-real) roots. If \(D = 0\), there is exactly one real, rational root (or two equal real roots). Since we calculated \(D = 0\), the equation has one real, rational solution. Dividing our original equation \(4x^2 - 32x + 64 = 0\) by 4 gives \(x^2 - 8x + 16 = 0\), which factors as \((x-4)^2 = 0\), confirming the solution is \(x = 4\), which is a rational number. Thus, the correct description is that there is one real, rational solution.
Examples
In engineering and physics, the discriminant is frequently used to determine the stability of systems or the intersection of paths. For instance, if an object's trajectory is modeled by a quadratic equation, finding the discriminant helps engineers determine if the object will hit a surface (two real roots), graze it at exactly one point (one real root), or miss it entirely (two complex roots). Understanding these properties allows for precise design and safety calculations in fields like ballistics and structural engineering.