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Algebra 1B Chapter 9. Solving Quadratic Equations The Quadratic Formula. Warm Up Evaluate for x = –2, y = 3, and z = –1. 1. x 2. 4. 2. xyz. 6. 3. x 2 – yz. 4. y – xz. 7. 1. 5. – x. 6. z 2 – xy. 7. 2. - PowerPoint PPT Presentation
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Algebra 1BAlgebra 1BChapter 9Chapter 9
Solving Quadratic EquationsThe Quadratic Formula
Warm Up
Evaluate for x = –2, y = 3, and z = –1.
6 1. x2 2. xyz
3. x2 – yz 4. y – xz
4
5. –x 6. z2 – xy
7 1
7 2
In the previous lesson, you completed the square to solve quadratic equations. If you complete the square of ax2 + bx + c = 0, you can derive the Quadratic Formula.
To add fractions, you need a common denominator.
Remember!
Additional Example 1A: Using the Quadratic Formula
Solve using the Quadratic Formula.
6x2 + 5x – 4 = 0
6x2 + 5x + (–4) = 0 Identify a, b, and c.
Use the Quadratic Formula.
Simplify.
Substitute 6 for a, 5 for b, and –4 for c.
Additional Example 1A Continued
Solve using the Quadratic Formula.
6x2 + 5x – 4 = 0
Simplify.
Write as two equations.
Solve each equation.
Additional Example 1B: Using the Quadratic Formula
Solve using the Quadratic Formula.
x2 = x + 20
1x2 + (–1x) + (–20) = 0 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.
Additional Example 1B Continued
Solve using the Quadratic Formula.
x = 5 or x = –4
Simplify.
Write as two equations.
Solve each equation.
x2 = x + 20
In Your Notes! Example 1a Solve using the Quadratic Formula. Check your answer.
–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
In Your Notes! Example 1a Continued
Solve using the Quadratic Formula. Check your answer.
Simplify.
Write as two equations.
Solve each equation.x = – or x = 2
–3x2 + 5x + 2 = 0
In Your Notes! Example 1b
Solve using the Quadratic Formula. Check your answer.
2 – 5x2 = –9x
Write in standard form. Identify a, b, and c.
(–5)x2 + 9x + (2) = 0
Use the Quadratic Formula.
Substitute –5 for a, 9 for b, and 2 for c.
Simplify
In Your Notes! Example 1b Continued
Solve using the Quadratic Formula. Check your answer.
Simplify.
Write as two equations.
Solve each equation.
2 – 5x2 = –9x
x = – or x = 2
In Your Notes! Example 1b Continued
Solve using the Quadratic Formula. Check your answer.
–5(2)2 + 9(2) + 2 0
–20 + 18 + 2 0
0 0
Check –5x2 + 9x + 2 = 0–5x2 + 9x + 2 = 0
–5 + 9 + 2 0
0 0
+ 2 0
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 x2 + 3x – 7 = 0 using the Quadratic Formula.
Check reasonableness
Estimate : x ≈ 1.54 or x ≈ –4.54.
In Your Notes! Example 2
Solve 2x2 – 8x + 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 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 x2 – 9x + 20 = 0. Show your work. Use at least two different methods. Check your answer.
Method 1 Solve by graphing.
y = x2 – 9x + 20Write the related quadratic
function and graph it.
The solutions are the x-intercepts, 4 and 5.
Additional Example 3 Continued
Solve x2 – 9x + 20 = 0. Show your work. Use at least two different methods. Check your answer.
Method 2 Solve by factoring.
x2 – 9x + 20 = 0
(x – 4)(x – 5) = 0
x – 4 = 0 or x – 5 = 0
x = 4 or x = 5
Factor.
Use the Zero Product Property.
Solve each equation.
Additional Example 3 Continued
Solve x2 – 9x + 20 = 0. Show your work. Use at least two different methods. Check your answer.
Check: 4 and 5
Check x2 – 9x + 20 = 0x2 – 9x + 20 = 0
(4)2 – 9(4) + 20 0 16 – 36 + 20 0
0 0
(5)2 – 9(5) + 20 0
25 – 45 + 20 0
0 0
In Your Notes! Example 3a
Solve. Show your work and check your answer.
x2 + 7x + 10 = 0
Method 3 Solve by completing the square.
x2 + 7x + 10 = 0
x2 + 7x = –10
x2 +7x = –10 Add to both sides.
Factor and simplify.
Take the square root of both sides.
In Your Notes! Example 3a Continued
Solve. Show your work and check your answer.
x2 + 7x + 10 = 0
Solve each equation.or
x = –2 or x = –5
(–2)2 + 7(–2) + 10 0
4 – 14 + 10 00 0
(–5)2 + 7(–5) + 10 0
25 – 35 + 10 0
0 0
Check x2 + 7x + 10 = 0 x2 + 7x + 10 = 0
In Your Notes! Example 3b
Solve. Show your work and check your answer.
–14 + x2 – 5x = 0
Method 4 Solve using the Quadratic Formula.
x2 – 5x – 14 = 0
1x2 – 5x – 14 = 0 Identify a, b, and c.
Substitute 1 for a, –5 for b, and –14 for c.
Simplify.
In Your Notes! Example 3b Continued
Solve. Show your work and check your answer.
–14 + x2 – 5x = 0
x = 7 or x = –2
or Write as two equations.
Solve each equation.
In Your Notes! Example 3b Continued
Solve. Show your work and check your answer.
–14 + x2 – 5x = 0
x2 – 5x – 14 = 0
72 – 5(7) – 14 0
49 – 35 – 14 0
14 – 14 0
0 0
Check x2 – 5x – 14 = 0
–22 – 5(–2) – 14 0
4 + 10 – 14 0
14 – 14 0
0 0
In Your Notes! Example 3c Solve. Show your work and check your answer.
2x2 + 4x – 21 = 0
Method 1 Solve by graphing.
2x2 + 4x – 21 = y Write the related quadratic 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 gives some advantages and disadvantages of the different methods.
