4
To determine the total number of quadratic equations that remain unchanged when their roots are squared, we need to consider what it means for a quadratic equation to have this property.
A quadratic equation is generally of the form:
\(ax^2 + bx + c = 0\)
In this problem, the highest degree coefficient (\(a\)) is 1, hence the equation simplifies to:
\(x^2 + bx + c = 0\)
Let \(r_1\) and \(r_2\) be the roots of this quadratic equation. The given condition implies that when these roots are squared, the equation formed still has \(x^2 + bx + c = 0\).
For a quadratic equation to remain unchanged on squaring its roots, we require that the squared roots satisfy the original equation's form. Let's explore how this works:
- Using Vieta's formulas, the sum and product of the roots \(r_1\) and \(r_2\) are given by:
- Sum of roots: \(r_1 + r_2 = -b\)
- Product of roots: \(r_1 \cdot r_2 = c\)
- When the roots are squared, say the new roots are \(r_1^2\) and \(r_2^2\), we have new sum and product as:
- Sum of squared roots: \(r_1^2 + r_2^2\)
- Product of squared roots: \(r_1^2 \cdot r_2^2 = (r_1 \cdot r_2)^2 = c^2\)
For the quadratic equation with squared roots to remain unchanged, the new sum and product must satisfy the equation \(x^2 + bx + c = 0\). Therefore:
- \(r_1^2 + r_2^2 = -b\)
- \((r_1 \cdot r_2)^2 = c^2\)
Now let's solve for possible integers (because quadratic polynomials have rational coefficients which suggest rational roots that can translate to perfect squares when squared):
- If the roots are 0 and 0, the equation is \(x^2\) and remains unchanged by squaring its roots.
- If the roots are 1 and 1 or -1 and -1, the equation is \((x - 1)^2 = x^2 - 2x + 1\) and \((x + 1)^2 = x^2 + 2x + 1\) respectively, both remain unchanged.
- If the roots are -1 and 1, the equation is \(x^2 - 1\) and remains unchanged.
Thus, we can see all possible equations with integer roots which satisfy the condition:
- \(x^2 - 1 = 0\)
- \(x^2 - 2x + 1 = 0\)
- \(x^2 + 2x + 1 = 0\)
- \(x^2 = 0\)
The total number of these quadratic equations is 4.
Therefore, the correct answer is:
Option: 4
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