Which quadratic equation has exactly one real solution (i.e., a repeated real root)?

• A. $2x^2+3x+1=0$
• B. $x^2+4x+3=0$
• C. $x^2+4x+5=0$
• D. $x^2+4x+4=0$

Answer: (D)

Verified step by step guidance

1. Recall that a quadratic equation of the form \(a x^{2} + b x + c = 0\) has exactly one real solution if and only if its discriminant is zero. The discriminant \(\Delta\) is given by the formula \(\Delta = b^{2} - 4 a c\).
2. For each quadratic equation, identify the coefficients \(a\), \(b\), and \(c\). For example, in \$2 x^{2} + 3 x + 1 = 0\(, \)a=2\(, \)b=3\(, and \)c=1$.
3. Calculate the discriminant for each quadratic using \(\Delta = b^{2} - 4 a c\). This will tell you the nature of the roots: if \(\Delta > 0\), there are two distinct real roots; if \(\Delta = 0\), there is exactly one real root (a repeated root); if \(\Delta < 0\), there are no real roots (complex roots).
4. Compare the discriminants of the given quadratics to find which one has \(\Delta = 0\). The quadratic with a zero discriminant is the one with exactly one real solution.
5. Verify that the quadratic \(x^{2} + 4 x + 4 = 0\) has \(a=1\), \(b=4\), and \(c=4\), and calculate its discriminant to confirm it equals zero, indicating a repeated real root.