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Solving Quadratic-Equation.pptx

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Susan Palacio
Susan Palacio

The document provides information about an online math class, including: - A prayer asking God for guidance and wisdom as students wait to be taught. - Reminders for online class such as turning on cameras, being on time, muting/unmuting audio. - The weekly task of answering pretest questions from the module. - An overview of the math topic for the first week of quarter 1 - quadratic equations. It discusses methods for solving quadratic equations such as extracting square roots, factoring, completing the square, and using the quadratic formula.

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Solving Quadratic-Equation.pptx
Dear God, we humbly gather together virtually. Thinking of starting today's online class. Oh, father, we come to you asking for your guidance, wisdom, and support as we wait to be pumped with knowledge by our teachers. Please help us understand whatever we are being taught, so we can have a brighter future when we finish our program. We pray this trusting your holy name. Amen.
REMINDERS 2.Always turn on your camera. 1.Be on time. 3.Unmute your audio when called by your teacher. 4.Do have your module, notebook and pen. 6.Be attentive and enjoy learning. 5.Questions will be entertained after a discussed topic.
ATTENDANCE
WEEKLY TASK SEPT. 3, 2022 (Saturday) Answer the Pretest Item # 1-5 on pp. 1-2 of your Module
MATH 9- QUARTER 1 - WEEK 1 QUADRATIC EQUATIONS
PIC -a- QUAD
Extracting Square Roots
Factoring
Completing the Square
Using the Quadratic
REVIEW: DEFINITIONS: A quadratic equation in one variable is a mathematical sentence of degree 2 that can be written in the standard form ax2 + bx + c = 0, where ,, and are real numbers and a 0. The values of ,, and are the numerical coefficients of quadratic term, linear term, and constant term, respectively. ax2 + bx + c = 0 Constant term Linear term Quadratic term
LESSON 2: SOLVING QUADRATIC EQUATIONS The solution set or roots of quadratic equations are values of the variable that will satisfy the equation.
METHODS USED TO SOLVE QUADRATIC EQUATIONS 1. Extracting Square Roots 2. Factoring 3. Completing the Square 4. Quadratic Formula
1. EXTRACTING SQUARE ROOTS/SQUARE ROOT PROPERTY
Example 1 x2 100 = 0 x = 賊 10 3x2 + 15 = 0 x2 100 + 100 = 0 + 100 x2 = 100 Solution Set: {10, -10} Example 5 Example 4 Example 3 Example 2 x2 = 0 x = 0 Solution Set: {0} 3x2 + 15 (-15) = 0 + (-15) 3x2 = -15 x2 = -5 Solution Set: No Solution 2x2 - 6 = 0 2x2 = 6 x2 = 3 (x 5)2 = 16 x - 5 = 賊4 x 5 + 5=賊4 +5 x = 5 賊 4 Solution Set: {9, 1} x = 5 + 4 x = 9 x = 1 x = 5 - 4
2. FACTORING Factoring is typically one of the easiest and quickest ways to solve quadratic equations; however, Not all quadratic polynomials can be factored. Applicable whenever the polynomial in the equation is factorable. This means that factoring will not work to solve many quadratic equations.
EXAMPLE 1: STEPS Solve: 2 = ( 5) = = 0 = 0 5 = 0 = 5 or The roots are 0 and 5. Transform the equation in standard form if necessary Factor the Polynomial Apply the Zero Product Property by setting each factor equal to zero Solve the equation
EXAMPLE 2: STEPS Solve: 2 = 9 - 18 2 9x + 18 = 9x 9x -18 + 18 - 6 = 0 = 6 3 = 0 = 3 or The roots are 6 and Transform the equation in standard form if necessary Write the equation in Standard Form by adding -9x and 18 on both sides. Apply the Zero Product Property by setting each factor equal to zero Solve the equation Simplify by combining like terms. 2 9 + 18 = 0 ( 6) ( 3) = 0 Factor the Polynomial.
3. COMPLETING THE SQUARE This method will work to solve ALL quadratic equations; however, it is messy to solve quadratic equations by completing the square if a 1 and/or b is an odd number. Completing the square is a great choice for solving quadratic equations if a = 1 and b is an even number.
3. COMPLETING THE SQUARE Before we use this method for solving quadratic equation, let us recall that if the expression 2 + + is a perfect square trinomial, it can be factored out into two identical binomial factors. 2 + 4 + 4 2 - 10 + 25 4 2 + 12 + 9 = ( + 2)( + 2) = ( + 2) 2 = (2 + 3) 2 = ( - 5) 2 = ( - 5)( - 5) = (2 + 3)(2 + 3)
EXAMPLE 1 X2 + 12X = 0 x2 + 12x = 0 x2 + 12x + ___ = 0 + ___ b= 12 12 / 2 =6 62=36 x2 + 12x + 36 = 0 + 36 (x + 6) 2 = 36 (x + 6) 2 = 36 x + 6 = +6 x + 6 = +6 x + 6 = -6 -6 = -6 - 6 = -6 x = 0 x = -12 The roots are 0 and -12. 1. Make one side a perfect square. Move the quadratic term and linear term to the left side of the equation 2. Add a blank to both sides 3. Divide b by 2 4. Square that answer. 5. Add it to both sides 6. Factor 1st side 7. Square root both sides 8. Solve for x
EXAMPLE 2 X2 + 6X + 8 = 0 x2 + 6x + 8 = 0 x2 + 6x + 8 -8 = 0 - 8 x2 + 6x + ___ = -8 + ___ b= 6 6 / 2 =3 32=9 x2 + 6x + 9 = -8 + 9 (x + 3) 2 = 1 (x + 3) 2 = 1 x + 3 = +1 x + 3 = +1 x + 6 = -1 - 3 = -3 - 3 = -3 x = -2 x = -4 The roots are -2 and -4. 1. Move constant to other side. Move the quadratic term and linear term to the left side of the equation 2. Add a blank to both sides 3. Divide b by 2 4. Square that answer. 5. Add it to both sides 6. Factor 1st side 7. Square root both sides 8. Solve for x
4. QUADRATIC FORMULA This method will work to solve ALL quadratic equations; however, for many equations it takes longer than some of the methods discussed earlier. The quadratic formula is a good choice if the quadratic polynomial cannot be factored, the equation cannot be written as (x+c)2 = n, or a is not 1 and/or b is an odd number.
Solve using the Quadratic Formula. 1.) x2 = x + 20 1x2 + (1x) + (20) = 0 a=1 b=-1 c=-20 Write in standard form. Identify a, b, and c. Use the quadratic formula. Simplify. Substitute 1 for a, 1 for b, and 20 for c.
Solve using the Quadratic Formula. x = 5 or x = 4 Simplify. Write as two equations. Solve each equation. x2 = x + 20
Solve using the Quadratic Formula. 2.) 3x2 + 5x + 2 = 0 Identify a, b, and c. Use the Quadratic Formula. Substitute 3 for a, 5 for b, and 2 for c. Simplify. 3x2 + 5x + 2 = 0 a=-3 b=5 c=2
Solve using the Quadratic Formula. Simplify. Write as two equations. Solve each equation. x = or x = 2 3x2 + 5x + 2 = 0
Solve using the Quadratic Formula. 3.) 2 5x2 = 9x Write in standard form. Identify a, b, and c. (5)x2 + 9x + (2) = 0 a=-5 b=9 c=2 Use the Quadratic Formula. Substitute 5 for a, 9 for b, and 2 for c. Simplify.
Method Most appropriate to use Extracting Square Roots or Square- Root Property for quadratic equations of the form 2 = Factoring If the constant term is 0, or if 2 + + is factorable. Completing the square For any quadratic equation of the form 2 + + , where a is 1 and b is even. Quadratic formula For any quadratic equation. REMEMBER
EXERCISES: SOLVE THE FOLLOWING USING THE APPROPRIATE METHOD OF SOLVING QUADRATIC EQUATIONS. 1. 2. 3. 4. 5. 2x2 2x + 3 = 0 x2 + 8x - 84 = 0 x2 = 121 x2 - 4x - 5 = 0 2x2 = 5x + 10
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Solving Quadratic-Equation.pptx

  • 2. Dear God, we humbly gather together virtually. Thinking of starting today's online class. Oh, father, we come to you asking for your guidance, wisdom, and support as we wait to be pumped with knowledge by our teachers. Please help us understand whatever we are being taught, so we can have a brighter future when we finish our program. We pray this trusting your holy name. Amen.
  • 3. REMINDERS 2.Always turn on your camera. 1.Be on time. 3.Unmute your audio when called by your teacher. 4.Do have your module, notebook and pen. 6.Be attentive and enjoy learning. 5.Questions will be entertained after a discussed topic.
  • 5. WEEKLY TASK SEPT. 3, 2022 (Saturday) Answer the Pretest Item # 1-5 on pp. 1-2 of your Module
  • 6. MATH 9- QUARTER 1 - WEEK 1 QUADRATIC EQUATIONS
  • 12. REVIEW: DEFINITIONS: A quadratic equation in one variable is a mathematical sentence of degree 2 that can be written in the standard form ax2 + bx + c = 0, where ,, and are real numbers and a 0. The values of ,, and are the numerical coefficients of quadratic term, linear term, and constant term, respectively. ax2 + bx + c = 0 Constant term Linear term Quadratic term
  • 13. LESSON 2: SOLVING QUADRATIC EQUATIONS The solution set or roots of quadratic equations are values of the variable that will satisfy the equation.
  • 14. METHODS USED TO SOLVE QUADRATIC EQUATIONS 1. Extracting Square Roots 2. Factoring 3. Completing the Square 4. Quadratic Formula
  • 16. Example 1 x2 100 = 0 x = 賊 10 3x2 + 15 = 0 x2 100 + 100 = 0 + 100 x2 = 100 Solution Set: {10, -10} Example 5 Example 4 Example 3 Example 2 x2 = 0 x = 0 Solution Set: {0} 3x2 + 15 (-15) = 0 + (-15) 3x2 = -15 x2 = -5 Solution Set: No Solution 2x2 - 6 = 0 2x2 = 6 x2 = 3 (x 5)2 = 16 x - 5 = 賊4 x 5 + 5=賊4 +5 x = 5 賊 4 Solution Set: {9, 1} x = 5 + 4 x = 9 x = 1 x = 5 - 4
  • 17. 2. FACTORING Factoring is typically one of the easiest and quickest ways to solve quadratic equations; however, Not all quadratic polynomials can be factored. Applicable whenever the polynomial in the equation is factorable. This means that factoring will not work to solve many quadratic equations.
  • 18. EXAMPLE 1: STEPS Solve: 2 = ( 5) = = 0 = 0 5 = 0 = 5 or The roots are 0 and 5. Transform the equation in standard form if necessary Factor the Polynomial Apply the Zero Product Property by setting each factor equal to zero Solve the equation
  • 19. EXAMPLE 2: STEPS Solve: 2 = 9 - 18 2 9x + 18 = 9x 9x -18 + 18 - 6 = 0 = 6 3 = 0 = 3 or The roots are 6 and Transform the equation in standard form if necessary Write the equation in Standard Form by adding -9x and 18 on both sides. Apply the Zero Product Property by setting each factor equal to zero Solve the equation Simplify by combining like terms. 2 9 + 18 = 0 ( 6) ( 3) = 0 Factor the Polynomial.
  • 20. 3. COMPLETING THE SQUARE This method will work to solve ALL quadratic equations; however, it is messy to solve quadratic equations by completing the square if a 1 and/or b is an odd number. Completing the square is a great choice for solving quadratic equations if a = 1 and b is an even number.
  • 21. 3. COMPLETING THE SQUARE Before we use this method for solving quadratic equation, let us recall that if the expression 2 + + is a perfect square trinomial, it can be factored out into two identical binomial factors. 2 + 4 + 4 2 - 10 + 25 4 2 + 12 + 9 = ( + 2)( + 2) = ( + 2) 2 = (2 + 3) 2 = ( - 5) 2 = ( - 5)( - 5) = (2 + 3)(2 + 3)
  • 22. EXAMPLE 1 X2 + 12X = 0 x2 + 12x = 0 x2 + 12x + ___ = 0 + ___ b= 12 12 / 2 =6 62=36 x2 + 12x + 36 = 0 + 36 (x + 6) 2 = 36 (x + 6) 2 = 36 x + 6 = +6 x + 6 = +6 x + 6 = -6 -6 = -6 - 6 = -6 x = 0 x = -12 The roots are 0 and -12. 1. Make one side a perfect square. Move the quadratic term and linear term to the left side of the equation 2. Add a blank to both sides 3. Divide b by 2 4. Square that answer. 5. Add it to both sides 6. Factor 1st side 7. Square root both sides 8. Solve for x
  • 23. EXAMPLE 2 X2 + 6X + 8 = 0 x2 + 6x + 8 = 0 x2 + 6x + 8 -8 = 0 - 8 x2 + 6x + ___ = -8 + ___ b= 6 6 / 2 =3 32=9 x2 + 6x + 9 = -8 + 9 (x + 3) 2 = 1 (x + 3) 2 = 1 x + 3 = +1 x + 3 = +1 x + 6 = -1 - 3 = -3 - 3 = -3 x = -2 x = -4 The roots are -2 and -4. 1. Move constant to other side. Move the quadratic term and linear term to the left side of the equation 2. Add a blank to both sides 3. Divide b by 2 4. Square that answer. 5. Add it to both sides 6. Factor 1st side 7. Square root both sides 8. Solve for x
  • 24. 4. QUADRATIC FORMULA This method will work to solve ALL quadratic equations; however, for many equations it takes longer than some of the methods discussed earlier. The quadratic formula is a good choice if the quadratic polynomial cannot be factored, the equation cannot be written as (x+c)2 = n, or a is not 1 and/or b is an odd number.
  • 25. Solve using the Quadratic Formula. 1.) x2 = x + 20 1x2 + (1x) + (20) = 0 a=1 b=-1 c=-20 Write in standard form. Identify a, b, and c. Use the quadratic formula. Simplify. Substitute 1 for a, 1 for b, and 20 for c.
  • 26. Solve using the Quadratic Formula. x = 5 or x = 4 Simplify. Write as two equations. Solve each equation. x2 = x + 20
  • 27. Solve using the Quadratic Formula. 2.) 3x2 + 5x + 2 = 0 Identify a, b, and c. Use the Quadratic Formula. Substitute 3 for a, 5 for b, and 2 for c. Simplify. 3x2 + 5x + 2 = 0 a=-3 b=5 c=2
  • 28. Solve using the Quadratic Formula. Simplify. Write as two equations. Solve each equation. x = or x = 2 3x2 + 5x + 2 = 0
  • 29. Solve using the Quadratic Formula. 3.) 2 5x2 = 9x Write in standard form. Identify a, b, and c. (5)x2 + 9x + (2) = 0 a=-5 b=9 c=2 Use the Quadratic Formula. Substitute 5 for a, 9 for b, and 2 for c. Simplify.
  • 30. Method Most appropriate to use Extracting Square Roots or Square- Root Property for quadratic equations of the form 2 = Factoring If the constant term is 0, or if 2 + + is factorable. Completing the square For any quadratic equation of the form 2 + + , where a is 1 and b is even. Quadratic formula For any quadratic equation. REMEMBER
  • 31. EXERCISES: SOLVE THE FOLLOWING USING THE APPROPRIATE METHOD OF SOLVING QUADRATIC EQUATIONS. 1. 2. 3. 4. 5. 2x2 2x + 3 = 0 x2 + 8x - 84 = 0 x2 = 121 x2 - 4x - 5 = 0 2x2 = 5x + 10