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Roots of quadratic equations Investigating the relationships between the roots and the coefficients of quadratic equations.

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Investigating roots Solve each of the following quadratic equations a)x 2 + 7x + 12 = 0 b)x 2 – 5x + 6 = 0 c)x 2 + x – 20 = 0 d)2x 2 – 5x – 3 = 0 Write down the sum of the roots and the product of the roots. Roots of polynomial equations are usually denoted by Greek letters. For a quadratic equation we use alpha (α) & beta (β) Investigate roots (excel)

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Properties of the roots of polynomial equations ax 2 + bx + c = 0 a(x - α)(x - β) = 0a = 0 This gives the identity ax 2 + bx + c = a(x - α)(x - β) Multiplying out ax 2 + bx + c = a(x 2 – αx – βx + αβ) = ax 2 – aαx – aβx + aαβ = ax 2 – ax(α + β) + aαβ Equating coefficients b = – a(α + β)c = aαβ -b/a = α + βc/a = αβ

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Task Use the quadratic formula to prove the results from the previous slide. -b/a = α + βc/a = αβ

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Properties of the roots of polynomial equations Find a quadratic equation with roots 2 & -5 -b/a = α + βc/a = αβ -b/a = 2 + -5c/a = -5 × 2 -b/a = -3c/a = -10 Taking a = 1 gives b = 3 & c = -10 So x 2 + 3x – 10 = 0 Note: There are infinitely many solutions to this problem. Taking a = 2 would lead to the equation 2x 2 + 6x – 20 = 0 Taking a = 1 gives us the easiest solution. If b and c are fractions you might like to pick an appropriate value for a.

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Properties of the roots of polynomial equations The roots of the equation 3x 2 – 10x – 8 = 0 are α & β 1 – Find the values of α + β and αβ. α + β = -b/a = 10/3 αβ = c/a = -8/3 2 –Find the quadratic equation with roots 3α and 3β. The sum of the new roots is 3α + 3β = 3(α + β) = 3 × 10/3 = 10 The product of the new roots is 9αβ = -24 From this we get that 10 = -b/a & -24 = c/a Taking a = 1 gives b = -10 & c = -24 So the equation is x 2 – 10x – 24 = 0

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3 – Find the quadratic equation with roots α + 2 and β + 2 The sum of the new roots is α + β + 4 = 10/3 + 4 = 22/3 The product of the new roots is (α + 2)(β + 2) = αβ + 2α + 2β + 4 = αβ + 2(α + β) + 4 = -8/3 + 2(10/3) + 4 = 8 So 22/3 = -b/a & 8 = c/a To get rid of the fraction let a = 3, so b = -22 & c = 24 The equation is 3x 2 – 22x + 24 = 0 Properties of the roots of polynomial equations

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The roots of the equation x 2 – 7x + 15 = 0 are α and β. Find the quadratic equation with roots α 2 and β 2 α + β = 7 & αβ = 15 (α + β) 2 = 49& α 2 β 2 = 225 α 2 + 2αβ + β 2 = 49 α 2 + 30 + β 2 = 49 α 2 + β 2 = 19 So the equation is x 2 – 19x + 225 = 0

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