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A quadratic equation is a second-order polynomial equation in a single variable x ax2+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, ax2+bx+c=0 x2+bax=−ca (x+b2a2)=−ca+b24a2=b2−4ac4a2 x+b2a=±√b2−4ac2a Solving for x then gives x=−b±√b2−4ac2a 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:
Example 1 3x2+4x−5=0 Add 5 to both sides 3x2+4x=5 Divide both sides by 3 x2+43x=53 Add (2/3)2 to both sides x2+43x+(23)2=53+(23)2 Factor the trinomial on the left side and combine the fractions on the right side (x−23)2=199 Take square root of both sides x−23=±√199 Add 2/3 to both sides. x=23±√199 Example 23x2−4x+1=0
a = 3, b = -4 and c = 1
substituting the numbers to the quadratic formula we have: x=−(−4)±√(−4)2−4∗3∗12∗3 x=4±√16−126 x=4±√46=4±26 x1=1 and x2=13
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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