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Nature of Roots of a Quadratic Equation in the Form of \(a{{x}^{2}}+bx+c=0\)

Nature of Roots of a Quadratic Equation in the Form of \(a{{x}^{2}}+bx+c=0\)

IP 3

Discriminant \({{b}^{2}}-4ac>0\)

Shape of curve

  • \(a{{x}^{2}}+bx+c=0\) has two distinct real roots / two distinct real solutions.

  • The curve of \(y=a{{x}^{2}}+bx+c\) intersects the \(x\)-axis at two distinct points.

 

Discriminant \({{b}^{2}}-4ac=0\)

Shape of curve

  • \(a{{x}^{2}}+bx+c=0\) has two identical real roots / two identical real solutions.

  • The curve of \(y=a{{x}^{2}}+bx+c\) touches the \(x\)-axis at one point / the \(x\)-axis is a tangent to the curve.

  • The curve of \(y=a{{x}^{2}}+bx+c\) is such that \(y\) is always non-negative (\(a>0\)), or, \(y\) is always non-positive (\(a<0\)).

Discriminant \({{b}^{2}}-4ac<0\)

Shape of curve

  • \(a{{x}^{2}}+bx+c=0\) has no real root / no real solutions.

  • The curve of \(y=a{{x}^{2}}+bx+c\) does not intersect the \(x\)-axis.

  • The curve of \(y=a{{x}^{2}}+bx+c\) is such that \(y\) is always positive / above the \(x\)-axis (\(a>0\)), or,  \(y\) is always negative / below the \(x\)-axis (\(a<0\)).

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