Algebra 1 B Chapter 9 Solving Quadratic Equations

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Algebra 1 B Chapter 9 Solving Quadratic Equations The Quadratic Formula

Algebra 1 B Chapter 9 Solving Quadratic Equations The Quadratic Formula

Warm Up Evaluate for x = – 2, y = 3, and z =

Warm Up Evaluate for x = – 2, y = 3, and z = – 1. 1. x 2 4 2. xyz 6 3. x 2 – yz 7 4. y – xz 1 5. –x 2 6. z 2 – xy 7

In the previous lesson, you completed the square to solve quadratic equations. If you

In the previous lesson, you completed the square to solve quadratic equations. If you complete the square of ax 2 + bx + c = 0, you can derive the Quadratic Formula.

Remember! To add fractions, you need a common denominator.

Remember! To add fractions, you need a common denominator.

Additional Example 1 A: Using the Quadratic Formula Solve using the Quadratic Formula. 6

Additional Example 1 A: Using the Quadratic Formula Solve using the Quadratic Formula. 6 x 2 + 5 x – 4 = 0 6 x 2 + 5 x + (– 4) = 0 Identify a, b, and c. Use the Quadratic Formula. Substitute 6 for a, 5 for b, and – 4 for c. Simplify.

Additional Example 1 A Continued Solve using the Quadratic Formula. 6 x 2 +

Additional Example 1 A Continued Solve using the Quadratic Formula. 6 x 2 + 5 x – 4 = 0 Simplify. Write as two equations. Solve each equation.

Additional Example 1 B: Using the Quadratic Formula Solve using the Quadratic Formula. x

Additional Example 1 B: Using the Quadratic Formula Solve using the Quadratic Formula. x 2 = x + 20 1 x 2 + (– 1 x) + (– 20) = 0 Write in standard form. Identify a, b, and c. Use the Quadratic Formula. Substitute 1 for a, – 1 for b, and – 20 for c. Simplify.

Additional Example 1 B Continued Solve using the Quadratic Formula. x 2 = x

Additional Example 1 B Continued Solve using the Quadratic Formula. x 2 = x + 20 Simplify. Write as two equations. x=5 or x = – 4 Solve each equation.

In Your Notes! Example 1 a Solve using the Quadratic Formula. Check your answer.

In Your Notes! Example 1 a Solve using the Quadratic Formula. Check your answer. – 3 x 2 + 5 x + 2 = 0 Identify a, b, and c. Use the Quadratic Formula. Substitute – 3 for a, 5 for b, and 2 for c. Simplify.

In Your Notes! Example 1 a Continued Solve using the Quadratic Formula. Check your

In Your Notes! Example 1 a Continued Solve using the Quadratic Formula. Check your answer. – 3 x 2 + 5 x + 2 = 0 Simplify. Write as two equations. x=– or x=2 Solve each equation.

In Your Notes! Example 1 b Solve using the Quadratic Formula. Check your answer.

In Your Notes! Example 1 b Solve using the Quadratic Formula. Check your answer. 2 – 5 x 2 = – 9 x (– 5)x 2 + 9 x + (2) = 0 Write in standard form. Identify a, b, and c. Use the Quadratic Formula. Substitute – 5 for a, 9 for b, and 2 for c. Simplify

In Your Notes! Example 1 b Continued Solve using the Quadratic Formula. Check your

In Your Notes! Example 1 b Continued Solve using the Quadratic Formula. Check your answer. 2 – 5 x 2 = – 9 x Simplify. Write as two equations. x=– or x = 2 Solve each equation.

In Your Notes! Example 1 b Continued Solve using the Quadratic Formula. Check your

In Your Notes! Example 1 b Continued Solve using the Quadratic Formula. Check your answer. Check – 5 x 2 + 9 x + 2 = 0 – 5(2)2 + 9(2) + 2 – 20 + 18 + 2 0 0 – 5 x 2 + 9 x + 2 = 0 – 5 +9 +2 0 0 0

Because the Quadratic Formula contains a square root, the solutions may be irrational. You

Because the Quadratic Formula contains a square root, the solutions may be irrational. You can give the exact solution by leaving the square root in your answer, or you can approximate the solutions.

Additional Example 2: Using the Quadratic Formula to Estimate Solutions Solve x 2 +

Additional Example 2: Using the Quadratic Formula to Estimate Solutions Solve x 2 + 3 x – 7 = 0 using the Quadratic Formula. Check reasonableness Estimate : x ≈ 1. 54 or x ≈ – 4. 54.

In Your Notes! Example 2 Solve 2 x 2 – 8 x + 1

In Your Notes! Example 2 Solve 2 x 2 – 8 x + 1 = 0 using the Quadratic Formula. Check reasonableness Estimate : x ≈ 3. 87 or x ≈ 0. 13.

There is no one correct way to solve a quadratic equation. Many quadratic equations

There is no one correct way to solve a quadratic equation. Many quadratic equations can be solved using several different methods: • Graphing • Factoring • Completing the square • Square roots and using • the Quadratic Formula

Additional Example 3: Solving Using Different Methods Solve x 2 – 9 x +

Additional Example 3: Solving Using Different Methods Solve x 2 – 9 x + 20 = 0. Show your work. Use at least two different methods. Check your answer. Method 1 Solve by graphing. Write the related quadratic 2 y = x – 9 x + 20 function and graph it. The solutions are the xintercepts, 4 and 5.

Additional Example 3 Continued Solve x 2 – 9 x + 20 = 0.

Additional Example 3 Continued Solve x 2 – 9 x + 20 = 0. Show your work. Use at least two different methods. Check your answer. Method 2 Solve by factoring. x 2 – 9 x + 20 = 0 (x – 4)(x – 5) = 0 Factor. x – 4 = 0 or x – 5 = 0 Use the Zero Product Property. x = 4 or x = 5 Solve each equation.

Additional Example 3 Continued Solve x 2 – 9 x + 20 = 0.

Additional Example 3 Continued Solve x 2 – 9 x + 20 = 0. Show your work. Use at least two different methods. Check your answer. Check: 4 and 5 Check x 2 – 9 x + 20 = 0 (4)2 – 9(4) + 20 0 16 – 36 + 20 0 x 2 – 9 x + 20 = 0 (5)2 – 9(5) + 20 25 – 45 + 20 0 0

In Your Notes! Example 3 a Solve. Show your work and check your answer.

In Your Notes! Example 3 a Solve. Show your work and check your answer. x 2 + 7 x + 10 = 0 Method 3 Solve by completing the square. x 2 + 7 x + 10 = 0 x 2 + 7 x = – 10 x 2 +7 x = – 10 Add to both sides. Factor and simplify. Take the square root of both sides.

In Your Notes! Example 3 a Continued Solve. Show your work and check your

In Your Notes! Example 3 a Continued Solve. Show your work and check your answer. x 2 + 7 x + 10 = 0 or Solve each equation. x = – 2 or x = – 5 Check x 2 + 7 x + 10 = 0 (– 2)2 + 7(– 2) + 10 4 – 14 + 10 0 0 x 2 + 7 x + 10 = 0 (– 5)2 + 7(– 5) + 10 0 25 – 35 + 10 0

In Your Notes! Example 3 b Solve. Show your work and check your answer.

In Your Notes! Example 3 b Solve. Show your work and check your answer. – 14 + x 2 – 5 x = 0 Method 4 Solve using the Quadratic Formula. x 2 – 5 x – 14 = 0 1 x 2 – 5 x – 14 = 0 Identify a, b, and c. Substitute 1 for a, – 5 for b, and – 14 for c. Simplify.

In Your Notes! Example 3 b Continued Solve. Show your work and check your

In Your Notes! Example 3 b Continued Solve. Show your work and check your answer. – 14 + x 2 – 5 x = 0 x=7 or Write as two equations. or x = – 2 Solve each equation.

In Your Notes! Example 3 b Continued Solve. Show your work and check your

In Your Notes! Example 3 b Continued Solve. Show your work and check your answer. – 14 + x 2 – 5 x = 0 Check x 2 – 5 x – 14 = 0 72 – 5(7) – 14 0 – 22 – 5(– 2) – 14 0 49 – 35 – 14 0 4 + 10 – 14 0 14 – 14 0 0 0

In Your Notes! Example 3 c Solve. Show your work and check your answer.

In Your Notes! Example 3 c Solve. Show your work and check your answer. 2 x 2 + 4 x – 21 = 0 Method 1 Solve by graphing. Write the related quadratic 2 x 2 + 4 x – 21 = y function. Divide each term by 2 and graph. The solutions are the x-intercepts and appear to be ≈ 2. 4 and ≈ – 4. 4.

Sometimes one method is better for solving certain types of equations. The table below

Sometimes one method is better for solving certain types of equations. The table below gives some advantages and disadvantages of the different methods.