Title: Quadratic equation with a negative discriminant

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A quadratic equation with a negative discriminant

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The solutions of the equation \(ax^2+bx+c = 0\) are \(x_{1,2} = \frac{-b\pm\sqrt{b^2-4\cdot a\cdot c}}{2\cdot a}\).

The expression under the square root sign in the solution formula for a quadratic equation is called the discriminant of the quadratic equation.

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In mathematics, complex numbers provide the ability to solve any quadratic equation, even with a negative discriminant.

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\(x_{1,2} = \frac{-b\pm\sqrt{b^2-4\cdot a\cdot c}^{ }}{2\cdot a}\)

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The solutions of the equation \(ax^2+bx+c = 0\) are

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Example. Solve the equation \(x^2-4x+13 = 0\)

Solution:

\(\displaystyle x_{1,2}=\frac{-(-4)\pm\sqrt{(-4)^2-4\cdot1\cdot13}}{2\cdot1}=\frac{4\pm\sqrt{-36}}{2}\)

\(\displaystyle x_{1,2}=\frac{4\pm6i}{2}=\frac{4}{2}\pm\frac{6i}{2}=2\pm3i\)

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\(\sqrt{36\cdot(-1)}=\sqrt{36}\cdot\sqrt{-1}\)

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Practice

SEquation

Solve the equation \(x\cdot(x-2)+26 = 0\)

Q: Answer: \(x_{1}=\)[...], \(x_{2}=\)[...].
A: 1-5i/1+5i; 1+5i/1-5i
Feedback (0 to 100%): Solution: \(x\cdot(x-2)+26 = 0 \Rightarrow x^2-2x+26 \Rightarrow D = (-2)^2-4\cdot26 = 4-104 = -100 \\ x_{1,2} = \frac{2\pm\sqrt{-100}}{2} = \frac{2\pm 10i}{2}\Rightarrow x_1 = 1+5i, x_2 = -1-5i \)

SEquation

Solve the equation \(x^2+961 = 0\)

Q: Answer: \(x_{1}=\)[...], \(x_{2}=\)[...].
A: -31i/31i; 31i/-31i
Feedback (0 to 100%): Solution: \(\Rightarrow x^2= -961 \Rightarrow x_{1,2} = \sqrt{-961} \Rightarrow x_{1,2} = \pm31i\)

SEquation

Solve the equation \(x^2+6x+34 = 0\)

Q: Answer: \(x_{1}\) = [...], \(x_{2}\) = [...].
A: -3-5i/-3+5i; -3+5i/-3-5i
Feedback (0 to 100%): Solution: \(x^2+6x+34 = 0 \Rightarrow D = 6^2-4\cdot34 = 36-136 = -100 \\ x_{1,2} = \frac{-6\pm\sqrt{-100}}{2} = \frac{-6\pm 10i}{2}\Rightarrow x_1 = -3+5i, x_2 = -3-5i\\\)