QUADRATIC EQUATION A Quadratic Equation in Standard Form ( a , b , and c can have any value, except that a can't be 0.) The letters a , b and c are coefficients (you know those values) The letter " x " is the variable or unknown (you don't know it yet)

EXAMPLE:

EXAMPLE The name Quadratic comes from "quad" meaning square, because of x 2 (in other words x squared ). It can also be called an equation of degree 2

SOME MORE EXAMPLES:

SOME MORE EXAMPLES In this one a=2 , b=5 and c=3

EXAMPLES CONTINUED:

EXAMPLES CONTINUED This one is a little more tricky: Where is a ? In fact a=1 , because we don't usually write "1x 2 " b=-3 And where is c ? Well, c=0 , so is not shown.

Slide 6:

We can solve Quadratic equation by 3 methods: 1:BY FACTORISATION METHOD 2: BY COMPLETING THE SQUARE METHOD 3:BY QUADRATIC FORMULA

Solving Quadratic Equations by Factorisation method :

Solving Quadratic Equations by Factorisation method

Slide 8:

Solving Quadratic Equations by Factorisation This presentation is an introduction to solving quadratic equations by factorisation. The following idea is used when solving quadratics by factorisation. If the product of two numbers is 0 then one (or both) of the numbers must be 0. So if xy = 0 either x = 0 or y = 0 Considering some specific numbers: If 8 x x = 0 then x = 0 If y x 15 = 0 then y = 0

Ex1 and 2:

Ex1 and 2 Solving Quadratic Equations by Factorisation a x 2 + b x + c = 0 , a 0 Some quadratic equations can be solved by factorising and it is normal to try this method first before resorting to the other two methods discussed. The first step in solving is to rearrange them (if necessary) into the form shown above. x 2 = 4 x Example 1: Solve 6 x 2 = – 9 x Example 2: Solve x 2 – 4 x = 0 x ( x – 4) = 0 either x = 0 or x – 4 = 0 if x – 4 = 0 then x = 4 Solutions (roots) are x = 0 , x = 4 6 x 2 + 9 x = 0 3 x (2 x + 3) = 0 either 3 x = 0 or 2 x + 3 = 0 x = 0 or x = – 1½ rearrange factorise rearrange factorise

Ex 3 and 4:

Ex 3 and 4 Solving Quadratic Equations by Factorisation a x 2 + b x + c = 0 , a 0 Some quadratic equations can be solved by factorising and it is normal to try this method first before resorting to the other two methods discussed. 4 x 2 = 9 Example 3: Solve x 2 – x – 12 = 0 Example 4: Solve 4 x 2 – 9 = 0 (2 x + 3 ) (2 x – 3) = 0 ( Using the difference of 2 squares) rearrange factorise factorise if 2 x + 3 = 0 then x = – 1½ if 2 x – 3 = 0 then x = 1½ Solutions (roots) are x = +/ – 1½ ( x + 3)( x – 4) = 0 if x + 3 = 0 then x = – 3 if x – 4 = 0 then x = 4 Solutions (roots) are x = – 3 or 4 The first step in solving is to rearrange them (if necessary) into the form shown above.

Ex 5 and 6:

Ex 5 and 6 Solving Quadratic Equations by Factorisation a x 2 + b x + c = 0 , a 0 Some quadratic equations can be solved by factorising and it is normal to try this method first before resorting to the other two methods discussed. 9 x 2 = 1 Example 5: Solve 6 x 2 = 3 – 7 x Example 6: Solve 9 x 2 – 1 = 0 (3 x + 1 ) (3 x – 1) = 0 ( Using the difference of 2 squares) rearrange factorise factorise if 3 x + 1 = 0 then x = – 1/3 if 3 x – 1 = 0 then x = 1/3 Solutions (roots) are x = +/ – 1/3 ( 2 x + 3)(3 x – 1) = 0 if 2 x + 3 = 0 then x = – 1½ if 3 x – 1 = 0 then x = 1/3 Solutions (roots) are x = – 1½ or 1/3 rearrange 6 x 2 + 7 x – 3 = 0 The first step in solving is to rearrange them (if necessary) into the form shown above.

Solving Quadratic Equations by Completing the Square:

Solving Quadratic Equations by Completing the Square

Perfect Square Trinomials:

Perfect Square Trinomials Examples x 2 + 6x + 9 x 2 - 10x + 25 x 2 + 12x + 36

Creating a Perfect Square Trinomial:

Creating a Perfect Square Trinomial In the following perfect square trinomial, the constant term is missing. X 2 + 14x + ____ Find the constant term by squaring half the coefficient of the linear term. (14/2) 2 X 2 + 14x + 49

Perfect Square Trinomials:

Perfect Square Trinomials Create perfect square trinomials. x 2 + 20x + ___ x 2 - 4x + ___ x 2 + 5x + ___ 100 4 25/4

Solving Quadratic Equations by Completing the Square:

Solving Quadratic Equations by Completing the Square Solve the following equation by completing the square: Step 1: Move quadratic term, and linear term to left side of the equation

Solving Quadratic Equations by Completing the Square:

Solving Quadratic Equations by Completing the Square Step 2: Find the term that completes the square on the left side of the equation. Add that term to both sides.

Slide 18:

Solving Quadratic Equations by Completing the Square Step 3: Factor the perfect square trinomial on the left side of the equation. Simplify the right side of the equation.

Solving Quadratic Equations by Completing the Square:

Solving Quadratic Equations by Completing the Square Step 4: Take the square root of each side

Solving Quadratic Equations by Completing the Square:

Solving Quadratic Equations by Completing the Square Step 5: Set up the two possibilities and solve

Completing the Square-Example #2:

Completing the Square-Example #2 Solve the following equation by completing the square: Step 1: Move quadratic term, and linear term to left side of the equation, the constant to the right side of the equation.

Slide 22:

Solving Quadratic Equations by Completing the Square Step 2: Find the term that completes the square on the left side of the equation. Add that term to both sides. The quadratic coefficient must be equal to 1 before you complete the square, so you must divide all terms by the quadratic coefficient first.

Slide 23:

Solving Quadratic Equations by Completing the Square Step 3: Factor the perfect square trinomial on the left side of the equation. Simplify the right side of the equation.

Slide 24:

Solving Quadratic Equations by Completing the Square Step 4: Take the square root of each side

Solving Quadratic Equations by Quadratic formula :

Solving Quadratic Equations by Quadratic formula

The Quadratic Formula.:

The Quadratic Formula.

What Does The Formula Do ?:

What Does The Formula Do ? The Quadratic formula allows you to find the roots of a quadratic equation (if they exist) even if the quadratic equation does not factorise. The formula states that for a quadratic equation of the form : ax 2 + bx + c = 0 The roots of the quadratic equation are given by :

Slide 28:

Example 1 Use the quadratic formula to solve the equation : x 2 + 5x + 6= 0 Solution: x 2 + 5x + 6= 0 a = 1 b = 5 c = 6 x = - 2 or x = - 3 These are the roots of the equation.

Slide 29:

Example 2 Use the quadratic formula to solve the equation : 8x 2 + 2x - 3= 0 Solution: 8x 2 + 2x - 3= 0 a = 8 b = 2 c = -3 x = ½ or x = - ¾ These are the roots of the equation.

Slide 30:

Example 3 Use the quadratic formula to solve the equation : 8x 2 - 22x + 15= 0 Solution: 8x 2 - 22x + 15= 0 a = 8 b = -22 c = 15 x = 3/2 or x = 5/4 These are the roots of the equation.

Slide 31:

Example 4 Use the quadratic formula to solve for x to 2 d.p : 2x 2 +3x - 7= 0 Solution: 2x 2 + 3x – 7 = 0 a = 2 b = 3 c = - 7 x = 1.27 or x = - 2.77 These are the roots of the equation.

Thank you:

Thank you This project was made by: Aditya bhattar STANDARD : x-A ROLL NUMBER : 2 SUBMITTED : KISHORE SIR

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