Quadratic Equation

What does it mean?

Definitions:

A quadratic equation is a second-order polynomial equation in a single variable x $$ax^2 + bx + c = 0$$ with a ≠ 0 . Because it is a second-order polynomial equation, the fundamental theorem of algebra guarantees that it has at least one solution. The solution may be real or complex.

The roots x can be found by completing the square, $$ax^2 + bx + c = 0$$ $$x^2 + \frac{b}{a}x = - \frac{c}{a}$$ $$(x + \frac{b}{2a}^2) = - \frac{c}{a} + \frac{b^2}{4a^2} = \frac{b^2 - 4ac}{4a^2}$$ $$x + \frac{b}{2a} = \frac{\pm\sqrt{b^2 - 4ac}}{2a}$$ Solving for x then gives $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ Quadratic Equation -- from Wolfram MathWorld

The expression $$b² - 4ac$$ that appears in the quadratic formula under the square root plays an important role in solving quadratic equations. Because of its importance: $$b² - 4ac$$ is called the determinant of the quadratic equation $$ax² + bx + c = 0$$

There are three possible cases:

b² - 4ac > 0. In this case, the equation has two distinct real roots.
b² - 4ac = 0. In this case, the equation has one real root. (called a double root).
b² - 4ac < 0. In this case, the equation does not have real roots.

What does it look like?

Example 1 $$3x^2 + 4x - 5 = 0$$ Add 5 to both sides $$3x^2 + 4x = 5$$ Divide both sides by 3 $$x^2 + \frac{4}{3}x = \frac{5}{3}$$ Add (2/3)² to both sides $$x^2 + \frac{4}{3}x + (\frac{2}{3})^2 = \frac{5}{3} + (\frac{2}{3})^2$$ Factor the trinomial on the left side and combine the fractions on the right side $$(x - \frac{2}{3})^2 = \frac{19}{9}$$ Take square root of both sides $$x - \frac{2}{3} = \pm\sqrt{\frac{19}{9}}$$ Add 2/3 to both sides. $$x =\frac{2}{3} \pm\sqrt{\frac{19}{9}}$$ Example 2$$3x^2 - 4x + 1 = 0$$

a = 3, b = -4 and c = 1

substituting the numbers to the quadratic formula we have: $$x = \frac{- (-4)\pm\sqrt{(-4)^2 - 4 * 3 * 1}}{2 * 3}$$ $$x = \frac{4\pm\sqrt{16 - 12}}{6}$$ $$x = \frac{4\pm\sqrt4}{6} = \frac{4\pm2}{6}$$ $$x_1 = 1 \space and \space x_2 = \frac{1}{3}$$

You'll use it...

Calculating distance, height and time of moving objects.

In application involving areas of the objects.

In banking calculating loan rates and profits.

Other applications where the quadratic equation is critical are: grandfather clocks, rabbits, areas, singing, tax, architecture, sundials, stopping, electronics, micro-chips, fridges, sunflowers, acceleration, paper, planets, ballistics, shooting, jumping, asteroids, quantum theory, chaos, windows, tennis, badminton, flight, radio, pendulum, weather, falling, shower, differential equations, telescope, golf.

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