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

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

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

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But what about NASTY trinomials that dont factor? But what about NASTY trinomials that dont factor? Abu Ja'far Muhammad ibn Musa Al-Khwarizmi Abu Ja'far Muhammad ibn Musa Al-Khwarizmi Born: about 780 in Baghdad (Iraq) Born: about 780 in Baghdad (Iraq) Died: about 850 Died: about 850

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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 doesnt factor! 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 doesnt factor!

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Plug in your answers for x. If youre right, youll get y = 0.

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Remember: All the terms must be on one side BEFORE you use the quadratic formula. Example: Solve 3m 2 - 8 = 10m Example: Solve 3m 2 - 8 = 10m Solution: 3m 2 - 10m - 8 = 0 Solution: 3m 2 - 10m - 8 = 0 a = 3, b = -10, c = -8 a = 3, b = -10, c = -8

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Solve: 3x 2 = 7 - 2x Solve: 3x 2 = 7 - 2x Solution: 3x 2 + 2x - 7 = 0 Solution: 3x 2 + 2x - 7 = 0 a = 3, b = 2, c = -7 a = 3, b = 2, c = -7

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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. 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: Dont do your homework late at night. MORAL: Dont do your homework late at night.

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