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Solving Quadratic Equations by Finding Square Roots

Solving Quadratic Equations by Finding Square Roots. What is a square root?. If a number square ( b 2 ) = another number ( a ), then b is the square root of a. Example: If 3 2 = 9, then 3 is the square root of 9. Some basics… .

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Solving Quadratic Equations by Finding Square Roots

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  1. Solving Quadratic Equations by Finding Square Roots

  2. What is a square root? • If a number square (b2) = another number (a), then b is the square root of a. • Example: If 32 = 9, then 3 is the square root of 9

  3. Some basics… • All positive numbers have two square roots, one positive and one negative

  4. Find the square root of numbers

  5. Perfect Squares: Numbers whose square roots are integers or quotients of integers.

  6. Evaluate a Radical Expression

  7. Quadratic Equations • Standard form: ax2 + bx + c = 0 • a is the leading coefficient and cannot be equal to zero. • If the value of b were equal to zero, the equation becomes ax2 + c = 0. • We can solve equations is this form by taking the square root of both sides.

  8. Key Concepts • When x2 = d • If d > 0, then x2 = d has two solutions • If d = 0, then x2 = d has one solution • If d < 0, then x2 = d has no real solution

  9. Solving quadratics • Solve each equation. a. x2=4 b. x2=5 c. x2=0 d. x2=-1 x2=4 has two solutions, x = 2, x = -2 x2=5 has two solutions, x =√5, x =- √5 x2=0 has one solution, x = 0 x2=-1 has no real solution, x = i, x = -i

  10. Solve by rewriting equation • Solve 3x2 – 48 = 0 3x2 – 48 + 48 = 0 + 48 3x2 = 48 3x2 / 3 = 48 / 3 x2 = 16 After taking square root of both sides, x = ± 4

  11. Solving Quadratic Equations • Solve. 3(x-2)2=21 • Solve. 3 - 5x2 = -9 -3 -3 -5x2 = -12 -5 -5 x2 = 3 3 (x-2)2 = 7

  12. More Examples! 4. Solve. • Solve. 4x2-6=42 +6 +6 4x2=48 4 4 x2 = 12

  13. Equation of a falling object • When an object is dropped, the speed with which it falls continues to increase. Ignoring air resistance, its height h can be approximated by the falling object model. h is the height in feet above the ground t is the number of seconds the object has been falling s is the initial height from which the object was dropped

  14. Falling Objects! • Use h = -16t2 + h0 Height of the object after it has fallen Object’s initial height # of seconds after the object is dropped

  15. Example • The tallest building in the USA is in Chicago, Illinois. It is 1450 ft. tall. How long would it take a penny to drop from the top of the building to the ground?

  16. Application An engineering student is in an “egg dropping contest.” The goal is to create a container for an egg so it can be dropped from a height of 32 feet without breaking the egg. To the nearest tenth of a second, about how long will it take for the egg’s container to hit the ground? Assume there is no air resistance.

  17. The question asks to find the time it takes for the container to hit the ground. • Initial height (s) = 32 feet • Height when its ground (h) = 0 feet • Time it takes to hit ground (t) = unknown

  18. Substitute 0 = -16t2 + 32 -32 + 0 = -16t2 + 32 – 32 -32 = -16t2 -32 / -16 = -16t2 / -16 2 = t2 t = √2 seconds or approx. 1.4 seconds

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