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Intro: We already know the standard form of a quadratic equation is: y = a x 2 + b x + c

The Quadratic Formula. Intro: We already know the standard form of a quadratic equation is: y = a x 2 + b x + c The constants are: a , b, c The variables are: y, x. What it means. The ROOTS (or solutions ) of a polynomial are its x-intercepts

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Intro: We already know the standard form of a quadratic equation is: y = a x 2 + b x + c

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  1. The Quadratic Formula • Intro: We already know the standard form of a quadratic equation is: y = ax2 + bx + c • The constants are: a , b, c • The variables are: y, x

  2. What it means • The ROOTS (or solutions) of a polynomial are its x-intercepts • Recall: The x-intercepts occur where y = 0.

  3. The Easy Way • Example: Find the roots: y = x2 + x - 6 • Solution: Factoring: y = (x + 3)(x - 2) • 0 = (x + 3)(x - 2) • The roots are: • x = -3; x = 2

  4. The Challenge • But what about NASTYtrinomials that don’t factor? • Abu Ja'far Muhammad ibn Musa Al-Khwarizmi • Born: about 780 in Baghdad (Iraq) • Died: about 850

  5. The Formula • After centuries of work, mathematicians realized that as long as you know the coefficients, you can find the roots of the quadratic. Even if it doesn’t factor!

  6. Example

  7. Solution!

  8. Your turn

  9. Your turn! • Solve: 3x2 = 7 - 2x • Solution: 3x2 + 2x - 7 = 0 • a = 3, b = 2, c = -7

  10. Weird History! We use the quadratic formula to solve second degree equations. Mathematicians tried for 300 years to solve higher-degree equations until Niels Abel (top picture) proved that no formula can be used to solve all fifth-degree equations. He was 22! • Evariste Galois (bottom picture) showed that there is no universal formula for any equations higher than the fourth degree. When Galois was 20, he wrote in ONE NIGHT much of the basis for a new theory of solving equations. Sadly, he was killed in a duel the next day. • MORAL: Don’t do your homework late at night.

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