DUE TUES/WED Solve each equation (use the method provided) y = x2 – 2x + 5 (quadratic formula) Formula Use When: The other methods do not apply
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DUE TUES/WED Solve each equation (use the method provided) y = x2 – 2x + 5 (quadratic formula) Formula Use When: The other methods do not apply
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REVIEW FOR QUIZ! NAME: __________________________ DUE TUES/WED Solve each equation (use the method provided) 1. Square Roots Method (x + 3)2 + 2 = -10 2. Factor to solve. x2 - 2x - 15 = 0 3. Complete the Square. x2 - 8x + 3 = 0 4. Quadratic Formula. 6x2 + 2x + 1= 0
Solve each equation. 5. y = 2 (x - 3)2 + 8 (square roots) 6. y = 2x2 + x - 10 (factor and zero prod) 7. y = x2 - 14x + 1 (complete the square) 8. y = x2 - 2x + 5 (quadratic formula)
SOLVING: WHICH METHOD SHOULD YOU USE? Explain why! Equation A B C D 1 x2 + 4x + 3 = 0 Sq. Roots Factor/ZPP Complete Sq. Quad. Form 2 5x2 - 1 = 6 Sq. Roots Factor/ZPP Complete Sq. Quad. Form 3 x2 - 7x + 1 = 0 Sq. Roots Factor/ZPP Complete Sq. Quad. Form 4 x2 + 10x + 4 = 0 Sq. Roots Factor/ZPP Complete Sq. Quad. Form 5 x2 - 14x = 5 Sq. Roots Factor/ZPP Complete Sq. Quad. Form 6 5 - 3x2 = 20 Sq. Roots Factor/ZPP Complete Sq. Quad. Form 7 x2 + x = 10 Sq. Roots Factor/ZPP Complete Sq. Quad. Form 8 x2 - 4x - 12 = 0 Sq. Roots Factor/ZPP Complete Sq. Quad. Form
Solve: Choose which method is best =)
1. y = 2 (x + 2)2 + 24 2. y = x2 - 6x + 8 3. y =
2x 2 -5x-124. y = x2 - 12x + 1
Find the discriminant and determine the number and type of solutions. Discriminant Number and Type of Solutions 5. y = 3x2 - 3x + 2 6. y = x2 - 10x + 1 7. y = x2 - 4x + 4
REVIEW PACKET SECTION 1: FACTORING Factor Completely! 1. x2 7x 6 2. x2 1003. 4x2 81
4. 25p2
16p 5. m2 10m 21
6. y2 3y 18 7. x2 7x 12 8. 4x2 20x 24
9. 4x2 20x 25
10. 16a2 49b2
11. 16x2 6x 12. x2 2x 1 13. 16a2 81
14. 3x2 3x 36
15. x2 8x 16 16. 24z2
14z 5 17. 3m2 7m 2 18. 3x2 75
SECTION 2: SOLVING
Use the square root method. 1. 5x2 7 602. x2 16 0 3. 5x2 9 134
4.
2(x+3)2
12 4Factor
and use the zero product property. 5. (2x 8) (x 5) 0 6. x2 2x 1 0 7. x2 6x 0 8. 6x2 11x 10Complete
theSquare.
9. x2 4x 12 0 10. x2 2x 350 11. x2 6x 23
12. 4x2 8x 40
Use the
Quadratic
Formula.
13. x2 5x 6 0 14. 2x2 4x 3 0 15. 2x2 x 4 2 16. 10x2 9 xSECTION 3: GRAPHING Find the vertex of each quadratic function: 1. f(x) = (x+ 2)2 + 5 ( , ) 2. f(x) = -2x2 - 3 ( , ) 3. f(x) = (x - 1)2 ( , ) 4. f(x) = 5x2 ( , ) 5. f(x) = (x + 10) (x - 2) ( , ) 6. f(x) = x2 + 2x + 5 ( , ) 7. f(x) = 2 (x - 5) (x + 3) ( , ) 8. f(x) = 2x2 + 8x + 5 ( , )
9. y -3 (x - 1)2 + 10 Opens Up or Opens DownStretched,
Shrink,
Standard
10. y (x + 4)2 + 4 Opens Up or Opens DownStretched,
Shrink,
Standard 11. Name 3 synonyms for "solution": _______________, _______________, _______________ Graph. 12.
y=2x+5 2 -3 13. y=- 1 2 x+5 x-3 14. y=x 2 +4x-6Quick Questions. Choose either ANSWER A or ANSWER B. QUESTION ANSWER A ANSWER B 1 What is the form of the function: y = 2x2 + 3x + 2 Intercept Form Standard Form 2 What is the form of the function: y = 2(x + 3)2 - 10 Vertex Form Intercept Form 3 What is the form of the function: y = - (x + 3) (x - 8) Intercept Form Standard Form 4 What formula will find the x-coordinate of the vertex for standard form?
x= -b 2a x= b 2 25 What formula will find the x-coordinate of the vertex for intercept form?
x= p-q 2 x= p+q 26 What is the value of C that would complete the square: x2 - 4x + C 4 16 7 What is the a-value: y = 2x2 + 5x + 2 1 2 8 What type of polynomial is always prime? A binomial sum of squares A trinomial 9 What method would you use to solve the equation: y = (x + 3) (2x + 1) Zero Product Property Complete the Square 10 What method would you use to solve the equation: y = 4x2 + 10 Square Roots Method Quadratic Formula 11 What method would you use to solve the equation: y = x2 + 10x + 3 Square Roots Method Complete the Square 12 The discriminant is 24. How many solutions are there? 2 1 13 The discriminant is -10. How many solutions are there? 0 2 14 The discriminant is 0. How many solutions are there? 1 0 15 The discriminant is -25. What type of solutions are there? Real Imaginary 16 The discriminant is 4. What type of solutions are there? Real Imaginary 17 How do you find any x-intercept? Substitute 0 for x Substitute 0 for y 18 How do you find any y-intercept? Substitute 0 for x Substitute 0 for y 19 What is the quadratic formula?
x= -bb 2 -4ac 2a x= -b±b 2 -4ac 2a20 What calculator function can you use to find the vertex of a parabola? 2nd Graph 2nd TRACE
WHAT SHOULD YOU DO NEXT? (when solving with square roots or factoring methods) 1. 2x2 + 8 = 10 A. Divide both sides by 2. B. Isolate x2. C. Square root both sides. 2. (x + 4)2 = 25 A. Distribute the square. B. FOIL. C. Square root both sides. 3. x2 - 25x = 0 A. Factor into (x + 5) (x - 5). B. Add 25x to both sides. C. Factor out x. 4. x2 + 5x + 4 = 0 A. Square root both sides. B. Subtract 4 from both sides. C. Factor the trinomial. 5. x2 + 3x = 10 A. Square root both sides. B. Subtract 3x from both sides. C. Subtract 10 from both sides. 6. (3x + 1) (x + 4) = 0 A. Set each factor equal to 0. B. FOIL. C. Combine like terms.
WHAT SHOULD YOU DO NEXT in order to factor?
7. 2x2 + 7x + 3 A. List pairs of factors of 3. B. Multiply 2 and 3. C. Factor out x. 8. 9x2 - 30x + 25 A. Try (3x - 5)2 and check it. B. Multiply 9 and 25. C. Set it equal to 0 and solve.
WRITE THE NEXT STEP ONLY! 1. (x + 5)2 = -49 2. x2 - 9x = 0 3. 2x2 + 4 = 8 4. 4x2 - 81 = 0 5. x - 2 =
±36. x + 1 =
±6i
7. Complete the square. x2 - 6x + 10 = 0 8. Complete the square. x2 + 8x = 3 9. Complete the square. x2 + 10x + 25 = 6 10. Quadratic Formula. x2 + 8x = 3 11. Quadratic Formula.
x=2±9-2(-2)(-4)
412. Quadratic Formula.
x= -10±6i2 4Unit 4 QUADRATICS Summary Sheet SECTION 1: FACTORING 1. Put the polynomial in order of decreasing degree (standard form). 10 + 7x + x2 All Types 2. Factor out the GCF (include any variables!) 4x2 + 14x Binomial A2 - B2 If it is a difference of squares, factor into conjugates. Formula: ___________________________________ x2 - 100 Binomial A2 + B2 If it is a sum of squares, the binomial is PRIME. x2 + 100 Trinomial x2 + Bx + C If A = 1, 1. List the pairs of factors of C. 2. Find a pair that has a sum/difference of the target #. 3. Write the two binomials. x2 + 7x + 12 Trinomial x2 + Bx + C If A = 1, 1. Multiply A and C and list pairs of factors. 2. Find a pair that has a sum/difference of the target #. 3. Factor by grouping. (or factor by trial and error)
2x2 - 3x - 20 Perfect Square Trinomial 1. If the first and last terms are perfect squares: 2. Try writing it as a binomial squared. 3. CHECK that the middle term works!! 4x2 + 28x + 49 SECTION 2: GRAPHING A quadratic function is a function with 2 as the highest degree (exponent) Vertex Form Intercept Form Standard Form
y=ax-h 2 +kVertex: (h, k) 1. a > 0: opens up a < 0: opens down 2. a < -1 or a > 1: stretched -1 < a < 1: compressed 3. Use the squares chart to find other points on the graph.
y=a(x-p)(x-q)Vertex:
p+q 2 , f p+q 21. Find the x-coordinate of the vertex. 2. Substitute it into the function to find the y-coordinate of the vertex. 3. Use the chart to find other points on the graph.
y=ax 2 +bx+cVertex:
-b 2a , f -b 2a1. Find the x-coordinate of the vertex. 2. Substitute it into the function to find the y-coordinate of the vertex. 3. Use the chart to find other points on the graph.
SECTION 3: SOLVING 1. Square Roots. Use When: An equation has an x2 or (x + c)2 (but does not have an x) 1. Isolate the x2. 2. Square root both sides. 3. Simplify (including the square root!) 4. Don't forget the ±
sign! 2. Factor and Zero Product Property. Use When: The equation is factorable. 1. Make sure the equation is in the form: ax2 + bx + c = 0 2. Factor completely! 3. Set each factor equal to 0. 4. Solve. 5. Write the solutions together: x = ____, ____ 3. Complete the Square. Use When: The trinomial is not factorable. A=1 and B is even. 1. Make sure the equation is in the form: Ax2 + Bx = C 2. Use the formula
B 2 2to determine C. 3. Add C to both sides. 4. Factor the left side of the equation into a binomial squared. 5. Take the square root of both sides (don't forget ±
) 6. Isolate the x. 4. Quadratic Formula. Use When: The other methods do not apply. 1. Put the equation into standard form: Ax2 + Bx + C = 0 2. Find A, B, C. 3. Substitute A, B, and C into the quadratic formula. Use parentheses! 4. Simplify completely! Quadratic Formula:
x= -b±b 2 -4ac 2aDiscriminant : b2 - 4ac If negative = 2 imaginary solutions If 0 = one real number solution If positive = 2 real number solutions Recall, i =
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