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Published byAngelica Elliott Modified over 4 years ago

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SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE BECAUSE GRAPHING IS SOMETIMES INACCURATE, ALGEBRA CAN BE USED TO FIND EXACT SOLUTIONS. ONE OF THOSE ALGEBRAIC METHODS IS “COMPLETING THE SQUARE”

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SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE Let’s solve x 2 – 10x + 18 = 0 Step 1: Get rid of constant on the left side x 2 – 10x + 18 = 0 -18 x 2 – 10x = -18

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SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE Let’s solve x 2 – 10x + 18 = 0 Step 2: Add constant to left side to create PST Half of middle term, then square it. x 2 – 10x = -18 + 25 Must add it to BOTH sides. + 25 (x – 5) 2 = 7

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SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE Let’s solve x 2 – 10x + 18 = 0 Step 3: Square root of both sides. (x – 5) 2 = 7

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SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE Let’s solve x 2 – 10x + 18 = 0 Step 4: Solve left side for x (x – 5) 2 = 7 +5

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SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE Try this one x 2 + 6x – 3 = 0 (x + 3) 2 = 12 -3 x 2 + 6x = 3 +3 Half of 6, squared +9 x 2 + 6x + 9 = 12

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COMPLETE THE SQUARE x 2 + 12x + _____ = 3 + _____ 36

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COMPLETE THE SQUARE x 2 – 8x + _____ = 10 + _____ 16

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COMPLETE THE SQUARE x 2 – 20x + _____ = 1 + _____ 100

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SOLVING QUADRATIC EQUATIONS USING THE QUADRATIC FORMULA Standard form for quadratic equations is ax 2 + bx + c = 0 and can be solved using the Quadratic Formula:

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SOLVING QUADRATIC EQUATIONS USING THE QUADRATIC FORMULA Example:3x 2 + 7x – 2 = 0

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THE DISCRIMINANT In a quadratic formula, the discriminant is the expression under the racical sign. What is the discriminant for 4x 2 + 2x – 7 = 0 ? b 2 – 4ac = 2 2 – 4(4)(-7)= 4 + 112 = 116

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THE DISCRIMINANT The discriminant tells you something about the roots of the equation. If the discriminant is negative, (b 2 – 4ac < 0), then there are no real roots (no solutions). If the discriminant is zero, (b 2 – 4ac = 0), then there is a double root (one solution). If the discriminant is positive, (b 2 – 4ac > 0), then there are two real roots.

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FLASH CARDS In the equation, x 2 + 5x – 6 = 0 a = 1

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FLASH CARDS In the equation, x 2 + 5x – 6 = 0 b = 5

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FLASH CARDS In the equation, x 2 + 5x – 6 = 0 c = -6

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FLASH CARDS In the equation, x 2 + 5x – 6 = 0 the discriminant = 49

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FLASH CARDS In the equation, 3x 2 – 6 = 0 the discriminant = 72

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FLASH CARDS How many roots if the discriminant is equal to 120 Two real roots

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FLASH CARDS How many roots if the discriminant is equal to 0 A double root

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FLASH CARDS How many roots if the discriminant is equal to 13 Two real roots

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FLASH CARDS How many roots if the discriminant is equal to -15 No real roots

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