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Solving Quadratic Equations

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Review of Solving Quadratic Equations ax 2 +bx +c = 0 A quadratic equation is a 2nd degree equation whose graph is a parabola.

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Solve with Perfect Square Binomial 1.Get Perfect Squares on Both Sides of Equation. 2.Take Square Root of Perfect Squares 3.Solve Positive Number & Neg. Number 4.Check Answers by Putting into Original Eq.

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Solve with Perfect Square Binomial 1.Get Perfect Squares Binomial. 2.Take Square Root of Both Sides of Eq 3.Simplify 4.Check Answers by Putting into Original Eq.

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Solve Quadratic Eq. By Factoring 1.Set Eq = 0 2.Factor 3.Set Each Factor = 0 4.Check Answers by Putting into Original Eq.

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What If You Can’t Factor The Equation? The “X” factor procedure doesn’t work which means it can not be factored -36 4

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You can solve this equation by Completing The Square

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Review of a Trinomial Square (x+5) 2 = x 2 + 10x + 25 When you square a binomial you get a TRINOMIAL SQUARE

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What is Completing The Square? x 2 + 10x + WHAT WILL MAKE A TRINOMIAL SQUARE? ?

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Now we will use these Algebra tiles to show Completing the Square 1 x x2x2 -x-x 2

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x 2 + 10x + ? It takes 25 ones to complete this square Note that 25 is half of b squared ( 10 / 2 ) 2 = 5 2 = 25 b

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2 2 Remember this to Complete The Square b

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Finding Half of b Squared b

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Add to Complete the Square b This is now a trinomial square

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x 2 +6x+9 is a Trinomial Square x 2 +6x+9 x+3 (x+3) 2 =x 2 +6x+9

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More Examples of Half of b Squared

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More Examples of Half of b Squared

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More Examples of Half of b Squared

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Complete The Square and State The Binomial Square

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Solve This Quad. Eq.

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If the “X” factor doesn’t work, you can’t solve by Factoring -31 4

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Solve by Completing The Square

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1.First Put c on One Side of equation.

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Solve by Completing The Square 1.First Put c on One Side of equation. 2.Divide by a Skip this step when a=1

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Solve by Completing The Square 1.First Put c on One Side of equation. 2.Divide by a 3.Complete The Square ADD HALF OF b SQUARED ( b / 2 ) 2 =( 4 / 2 ) 2 =4 b

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Solve by Completing The Square 1.First Put c on One Side of equation. 2.Divide by a 3.Complete The Square 4.Write Trinomial Square as a Binomial Square

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Solve by Completing The Square 1.First Put c on One Side of equation. 2.Divide by a 3.Complete The Square 4.Write Trinomial Square as a Binomial Square 5.Square Root both sides

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Solve by Completing The Square 1.First Put c on One Side of equation. 2.Divide by a 3.Complete The Square 4.Write Trinomial Square as a Binomial Square 5.Square Root both sides 6.Solve for x (Note 2 Answers)

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Solve by Completing The Square 1.First Put c on One Side of equation. 2.Divide equation by a 3.Complete The Square ( b / 2 ) 2 = 4.Write Trinomial Square as a Binomial Square 5.Square Root both sides 6.Solve for x (Note 2 Answers)

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Solve by Completing The Square 1.First Put c on One Side of equation. 2.Divide by a 3.Complete The Square ( b / 2 ) 2 =( 8 / 2 ) 2 =4 2 = 16 4.Write Trinomial Square as a Binomial Square 5.Square Root both sides 6.Solve for x (Note 2 Answers)

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Solve by Completing The Square 1.First Put c on One Side of equation. 2.Divide by a 3.Complete The Square ( b / 2 ) 2 = 4.Write Trinomial Square as a Binomial Square 5.Square Root both sides 6.Solve for x (Note 2 Answers)

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Solve by Completing The Square 1.First Put c on One Side of equation. 2.Divide by a 3.Complete The Square ( b / 2 ) 2 =( 2 / 2 ) 2 =1 4.Write Trinomial Square as a Binomial Square 5.Square Root both sides 6.Solve for x (Note 2 Answers)

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Solve by Completing The Square

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1.First Put c on One Side of equation. 2.Divide by a 3.Complete The Square ( b / 2 ) 2 = ( 6 / 2 ) 2 =3 2 =9 4.Write Trinomial Square as a Binomial Square 5.Square Root both sides 6.Solve for x (Note 2 Answers)

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Solve by Completing The Square 1.First Put c on One Side of equation. 2.Divide by a 3.Complete The Square ( b / 2 ) 2 = ( 9 / 2 ) 2 = 81 / 4 4.Write Trinomial Square as a Binomial Square 5.Square Root both sides 6.Solve for x (Note 2 Answers)

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Solve by Completing The Square 1.First Put c on One Side of equation. 2.Divide by a 3.Complete The Square ( b / 2 ) 2 = ( 2 / 2 ) 2 =1 4.Write Trinomial Square as a Binomial Square 5.Square Root both sides 6.Solve for x (Note 2 Answers)

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Solve Quadratic Equations With The Quadratic Formula So far we have solved quadratic equations with FACTORING (Solves Equations with Integer Solutions) & COMPLETING THE SQUARE (Solves All quadratic equations) THE QUADRATIC FORMULA ALSO SOLVES ALL QUADRATIC EQUATIONS

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QUADRATIC EQUATION

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- b 2 a + 2 X=

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Solve with Quadratic Equation

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1.Put Equation into Std. Form (ax 2 +bx+c=0) a=1bc

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Solve with Quadratic Equation 1.Put Equation into Std. Form (ax 2 +bx+c=0) 2.Plug a, b & c into the Quad. Eq. a=1bc

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Solve with Quadratic Equation 1.Put Equation into Std. Form (ax 2 +bx+c=0) 2.Plug a, b & c into the Quad. Eq. 3.Simplify Radical a=1bc

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Solve with Quadratic Equation 1.Put Equation into Std. Form (ax 2 +bx+c=0) 2.Plug a, b & c into the Quad. Eq. 3.Simplify Radical 4.Solve for x a=1bc

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Solve with Quadratic Equation

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1.Put Equation into Std. Form (ax 2 +bx+c=0) a=1bc

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Solve with Quadratic Equation 1.Put Equation into Std. Form (ax 2 +bx+c=0) 2.Plug a, b & c into the Quad. Eq. a=1bc

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Solve with Quadratic Equation 1.Put Equation into Std. Form (ax 2 +bx+c=0) 2.Plug a, b & c into the Quad. Eq. 3.Simplify Radical a=1bc

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Solve with Quadratic Equation 1.Put Equation into Std. Form (ax 2 +bx+c=0) 2.Plug a, b & c into the Quad. Eq. 3.Simplify Radical 4.Solve for x a=1bc

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Solve by quadratic equation

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Solve with Quadratic Equation 1.Put Equation into Std. Form (ax 2 +bx+c=0) 2.Plug a, b & c into the Quad. Eq. 3.Simplify Radical 4.Solve for x a=1bc

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Solve with Quadratic Equation 1.Put Equation into Std. Form (ax 2 +bx+c=0) 2.Plug a, b & c into the Quad. Eq. 3.Simplify Radical 4.Solve for x a=1bc

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Solve with Quadratic Equation 1.Put Equation into Std. Form (ax 2 +bx+c=0) 2.Plug a, b & c into the Quad. Eq. 3.Simplify Radical 4.Solve for x abc

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The Discriminant The Discriminant equals b 2 -4ac The Discriminant is what’s under the RADICAL

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The Discriminant The Discriminant equals b 2 -4ac 2 Solutions if positive 1 Solution if 0 0 Solutions if negative

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The Discriminant tells you how many solutions or roots x This quadratic has two roots so the Discriminant must be positive y

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The Discriminant tells you how many solutions or roots This quadratic has one roots so the Discriminant must be zero X

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The Discriminant tells you how many solutions or roots This quadratic has no roots so the Discriminant must be negative X

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Find The Number of Solutions (Roots) 1.Identify a, b & c. 2.Put values into the discriminant b 2 -4ac 3.If Pos. 2 Solutions If Zero 1 Solution If Neg. 0 Solutions The discriminant is neg. so there are no solutions

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Determine The Number of Solutions or Roots

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Singing the Quadratic Equation (To the melody of Pop Goes The Weasel) Minus B plus or minus the square root B squared minus 4AC All over 2A, That’s the quadratic equation. It’s the greatest equation of all time. It even has a wonderful rhyme. If you don’t know it, It would be a crime That’s the quadratic equation. Math is the most taught subject in the world So you’d better believe me Bone up to pass the SAT You won’t end up a weasel.

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SOLVING QUADRATIC EQUATIONS

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What Method To Use SquareRoot when there is a perfect square on one side of eq. Use “X” Factor when trinomial is factorable Use quadratic formula or complete the square if not factorable

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