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Math Skills Overview Guide

Quadratic Equation

What does it mean?

Definitions:

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=b24ac4a2 x+b2a=±b24ac2a Solving for x then gives x=b±b24ac2a Quadratic Equation -- from Wolfram MathWorld

 

The expression b²4ac that appears in the quadratic formula under the square root plays an important role in solving quadratic equations. Because of its importance: b²4ac is called the determinant of the quadratic equation ax²+bx+c=0

There are three possible cases:

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

 What does it look like?

Example 1 3x2+4x5=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 (x23)2=199 Take square root of both sides x23=±199 Add 2/3 to both sides. x=23±199 Example 23x24x+1=0

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

substituting the numbers to the quadratic formula we have: x=(4)±(4)243123 x=4±16126 x=4±46=4±26 x1=1 and x2=13

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.

From +Plus Magazine

  Video

Completing the Square

Watch a Khan Academy Video »
Length: 13:45

  Video

Quadratic Formula 1

Watch a Khan Academy Video »
Length: 5:35