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6.5 Quadratic Formula

Derivation of the quadratic formula. Solve the equation below by completing the square to derive the quadratic formulaax2 bx c = 0divide by ax2 (b/a)x (c/a) = 0 x2 (b/a)x = -c/a move (c/a) to rightNow complete the square

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6.5 Quadratic Formula

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    1. 6.5 Quadratic Formula Using the quadratic formula Discriminant Projectile motion formula

    2. Derivation of the quadratic formula Solve the equation below by completing the square to derive the quadratic formula ax2 + bx + c = 0 divide by a x2 + (b/a)x + (c/a) = 0 x2 + (b/a)x = -c/a move (c/a) to right Now complete the square – add (b/a * ˝)2 to each side (b/a * ˝)2 = (b/2a)2 = ?? b2/4a2

    3. Derivation, continued x2 + (b/a)x + (b2/4a2) = (-c/a) + (b2/4a2) Rewriting the left side as a perfect square, (x + b/2a)2 = (-c/a) + (b2/4a2) Now combine the right side by using 4a2 as a common denominator; Right side: (-c*4a / 4a2) + (b2/4a2) Next: (-4ac/4a2 + b2/4a2) = (b2 – 4ac)/4a2

    4. Derivation, concluded So the equation is now: (x + b/2a)2 = (b2 – 4ac)/4a2 Square root each side x + b/2a = (± sqroot (b2 – 4ac)) / 2a Finally, move (b/2a) from the left side .. Note that both fractions have denominator of 2a… so: x = (-b ± sqroot (b2 – 4ac)) / 2a

    5. Using the quadratic formula Make sure equation is in the form: ax2 + bx + c = 0 Substitute a,b, and c into the quadratic formula Be CAREFUL when simplifying, easy to make careless errors In abstract problems, generally express answers in simplified radical form if the answers are irrational.. In word problems, usually get a decimal approximation

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