Question:
\(a\), \(b\) and \(c\) are the sides of a triangle. The equations \(ax^2 + bx + c = 0\) and \(3x^2 + 4x + 5 = 0\) have a common root. Then angle C is equal to
\(a\), \(b\) and \(c\) are the sides of a triangle. The equations \(ax^2 + bx + c = 0\) and \(3x^2 + 4x + 5 = 0\) have a common root. Then angle C is equal to
Show Hint
The equation 3x^2+4x+5=0 has no real roots, so any real-coefficient quadratic sharing one of its roots must share both, making a, b, c proportional to 3, 4, 5, a Pythagorean triple.
Updated On: Jul 13, 2026
- \(60^{\circ}\)
- \(90^{\circ}\)
- \(120^{\circ}\)
- None of these
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The Correct Option is B
Solution and Explanation
Step 1: Check the nature of the roots of the fixed equation.
For \(3x^2+4x+5=0\), the discriminant is \(D = 4^2 - 4(3)(5) = 16 - 60 = -44\), which is negative. So this equation has no real roots, only a pair of complex conjugate roots.
Step 2: Use the fact that a, b, c are real.
Since a, b, c are the real sides of a triangle, the equation \(ax^2+bx+c=0\) has real coefficients. Whenever a quadratic with real coefficients has a non-real root, its complex conjugate must also be a root.
Step 3: See what sharing one root forces.
We are told the two equations share a common root. Since \(3x^2+4x+5=0\) has only a complex conjugate pair as its two roots, and \(ax^2+bx+c=0\) is also a quadratic with real coefficients, the moment they share one non-real root, they must share the conjugate too, because a real quadratic cannot have just one non-real root on its own. So the two quadratics have the exact same pair of roots.
Step 4: Two quadratics with identical roots must be proportional.
If \(ax^2+bx+c=0\) and \(3x^2+4x+5=0\) have identical roots, their coefficients are proportional:
\[ \frac{a}{3} = \frac{b}{4} = \frac{c}{5} = k \]
So \(a = 3k\), \(b = 4k\), \(c = 5k\) for some nonzero constant k.
Step 5: Recognize the triangle.
The sides are in the ratio 3:4:5, the well known Pythagorean triple, since \(3^2+4^2=5^2\). So the triangle is right angled, and the right angle sits opposite the longest side, \(c = 5k\).
Step 6: Identify angle C.
By convention, angle C is the angle opposite side c. Since c is the longest side here, angle C is the right angle.
Final Answer:
Angle C = 90°.
\[ \boxed{90^{\circ}} \]
For \(3x^2+4x+5=0\), the discriminant is \(D = 4^2 - 4(3)(5) = 16 - 60 = -44\), which is negative. So this equation has no real roots, only a pair of complex conjugate roots.
Step 2: Use the fact that a, b, c are real.
Since a, b, c are the real sides of a triangle, the equation \(ax^2+bx+c=0\) has real coefficients. Whenever a quadratic with real coefficients has a non-real root, its complex conjugate must also be a root.
Step 3: See what sharing one root forces.
We are told the two equations share a common root. Since \(3x^2+4x+5=0\) has only a complex conjugate pair as its two roots, and \(ax^2+bx+c=0\) is also a quadratic with real coefficients, the moment they share one non-real root, they must share the conjugate too, because a real quadratic cannot have just one non-real root on its own. So the two quadratics have the exact same pair of roots.
Step 4: Two quadratics with identical roots must be proportional.
If \(ax^2+bx+c=0\) and \(3x^2+4x+5=0\) have identical roots, their coefficients are proportional:
\[ \frac{a}{3} = \frac{b}{4} = \frac{c}{5} = k \]
So \(a = 3k\), \(b = 4k\), \(c = 5k\) for some nonzero constant k.
Step 5: Recognize the triangle.
The sides are in the ratio 3:4:5, the well known Pythagorean triple, since \(3^2+4^2=5^2\). So the triangle is right angled, and the right angle sits opposite the longest side, \(c = 5k\).
Step 6: Identify angle C.
By convention, angle C is the angle opposite side c. Since c is the longest side here, angle C is the right angle.
Final Answer:
Angle C = 90°.
\[ \boxed{90^{\circ}} \]
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