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Algebra 1B Chapter 9

Learn how to solve quadratic equations by graphing the related function, finding the vertex and zeros, and determining the real-number solutions. Examples provided.

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Algebra 1B Chapter 9

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  1. Algebra 1BChapter 9 Solving Quadratic Equations By Graphing

  2. Warm Up 1. Graph y = x2 + 4x + 3. 2. Identify the vertex and zeros of the function above. vertex:(–2 , –1); zeros:–3, –1

  3. Every quadratic function has a related quadratic equation. The standard form of a quadratic equation is ax2 + bx + c = 0, where a, b, and c are real numbers and a ≠ 0. When writing a quadratic function as its related quadratic equation, you replace y with 0. y=ax2 + bx + c 0 = ax2 + bx + c

  4. One way to solve a quadratic equation in standard form is to graph the related function and find the x-values where y = 0. In other words, find the zeros of the related function. Recall that a quadratic function may have two, one, or no zeros.

  5. Additional Example 1A: Solving Quadratic Equations by Graphing Solve the equation by graphing the related function. 2x2 – 18 = 0 Step 1 Write the related function. 2x2 – 18 = y,or y = 2x2+ 0x– 18 Step 2 Graph the function. ● • The axis of symmetry is x = 0. • The vertex is (0, –18). • Two other points (2, –10) and • (3, 0) • Graph the points and reflectthem • across the axis of symmetry. x = 0 ● (3, 0) ● ● (2, –10) ● (0, –18)

  6. Check2x2 – 18 = 0 2x2 – 18 = 0 2(3)2 – 18 0 2(–3)2 – 18 0 2(9) – 18 0 2(9) – 18 0 18 – 18 0 18 – 18 0   0 0 0 0 Additional Example 1A Continued Solve the equation by graphing the related function. 2x2 – 18 = 0 Step 3 Find the zeros. The zeros appear to be 3 and –3. The solutions of 2x2 – 18 = 0 are 3 and –3. Substitute 3 and –3 for x in the original equation.

  7. 2x2 – 12x + 18 = 0 y = 2x2 – 12x + 18 Additional Example 1B: Solving Quadratic Equations by Graphing Solve the equation by graphing the related function. –12x + 18 = –2x2 Step 1 Write the related function. Step 2 Graph the function. Use a graphing calculator. Step 3 Find the zeros. The only zero appears to be 3. This means 3 is the only root of 2x2 – 12x + 18.

  8. Step 1 Write the related function. y = 2x2 + 4x + 3  (–3, 9)  (1, 9)  (–2, 3)  (0, 3)  (–1, 1) Additional Example 1C: Solving Quadratic Equations by Graphing Solve the equation by graphing the related function. 2x2 + 4x = –3 Step 2 Graph the function. • The axis of symmetry is x = –1. • The vertex is (–1, 1). • Two other points (0, 3) and • (1, 9). • Graph the points and reflectthem • across the axis of symmetry.

  9. Additional Example 1C Continued Solve the equation by graphing the related function. 2x2 + 4x = –3 Step 3 Find the zeros. The function appears to have no zeros. The equation has no real-number solutions.

  10. In Your Notes! Example 1a Solve the equation by graphing the related function. x2 – 8x – 16 = 2x2 Step 1 Write the related function. y = x2 + 8x+ 16 x = –4 Step 2 Graph the function. • The axis of symmetry is x = –4. • The vertex is (–4, 0). • The y-intercept is 16. • Two other points are (–3, 1) and • (–2, 4). • Graph the points and reflectthem • across the axis of symmetry. ● ● (–2 , 4) ● ● (–3, 1) ● (–4, 0)

  11. Check y =x2 + 8x + 16 0 (–4)2 + 8(–4) + 16 0 16 – 32 + 16  0 0 In Your Notes! Example 1a Continued Solve the equation by graphing the related function. x2 – 8x – 16 = 2x2 Step 3 Find the zeros. The only zero appears to be –4. Substitute –4 for x in the quadratic equation.

  12. Step 1 Write the related function. y =x2 + 6x + 10 In Your Notes! Example 1b Solve the equation by graphing the related function. 6x + 10 = –x2 x = –3 Step 2 Graph the function. • The axis of symmetry is x = –3 . • The vertex is (–3 , 1). • The y-intercept is 10. • Two other points (–1, 5) and • (–2, 2) • Graph the points and reflectthem • across the axis of symmetry. (–1, 5) ● ● (–2, 2) ● ● ● (–3, 1)

  13. Step 3 Find the zeros. The function appears to have no zeros In Your Notes! Example 1b Continued Solve the equation by graphing the related function. x2 + 6x + 10 = 0 The equation has no real-number solutions.

  14. y = –x2 + 4 Step 2 Graph the function. Use a graphing calculator. In Your Notes! Example 1c Solve the equation by graphing the related function. –x2 + 4 = 0 Step 1 Write the related function. Step 3 Find the zeros. The function appears to have zeros at (2, 0) and (–2, 0).

  15. Finding the roots of a quadratic polynomial is the same as solving the related quadratic equation.

  16.    Additional Example 2A: Finding Roots of Quadratic Polynomials Find the roots of x2 + 4x + 3 Step 1 Write the related equation. 0 = x2 + 4x + 3 y = x2 + 4x + 3 Step 2 Write the related function. y = x2 + 4x + 3 Step 3 Graph the related function. (–4, 3) • The axis of symmetry is x = –2. • The vertex is (–2, –1). • Two other points are (–3, 0) • and (–4, 3) • Graph the points and reflectthem • across the axis of symmetry. (–3, 0)  (–2, –1)

  17. 0 = x2 + 4x + 3 Check 0 = x2 + 4x + 3 0(–3)2 + 4(–3) + 3 0(–1)2 + 4(–1) + 3 09 – 12 + 3 01 – 4 + 3   0 0 0 0 Additional Example 2A Continued Find the roots of each quadratic polynomial. Step 4 Find the zeros. The zeros appear to be –3 and –1. This means –3 and –1 are the roots of x2 + 4x + 3.

  18. (2, –15)     (1, –18)  (–0.5, –20.25). Additional Example 2B: Finding Roots of Quadratic Polynomials Find the roots of x2 + x – 20 Step 1 Write the related equation. y = x2 + 4x – 20 0 = x2 + x – 20 Step 2 Write the related function. y = x2 + 4x – 20 Step 3 Graph the related function. • The axis of symmetry is x = – . • The vertex is (–0.5, –20.25). • Two other points are (1, –18) • and (2, –15) • Graph the points and reflectthem • across the axis of symmetry.

  19. 0 = x2 + x – 20 Check 0 = x2 + x – 20 0(–5)2– 5 – 20 042 + 4 – 20 025 – 5 – 20 016 + 4 – 20   0 0 0 0 Additional Example 2B Continued Find the roots of x2 + x – 20 Step 4 Find the zeros. The zeros appear to be –5 and 4. This means –5 and 4 are the roots of x2 + x – 20.

  20. (4, 3)    (5, 0)   (6, –1). Additional Example 2C: Finding Roots of Quadratic Polynomials Find the roots of x2 – 12x + 35 Step 1 Write the related equation. y = x2 – 12x + 35 0 = x2 – 12x + 35 Step 2 Write the related function. y= x2 – 12x + 35 Step 3 Graph the related function. • The axis of symmetry is x = 6. • The vertex is (6, –1). • Two other points (4, 3) and • (5, 0) • Graph the points and reflectthem • across the axis of symmetry.

  21. 0 = x2 – 12x + 35 Check 0 = x2 – 12x + 35 052 – 12(5) + 35 072 – 12(7) + 35 025 – 60 + 35 049 – 84 + 35   0 0 0 0 Additional Example 2C Continued Find the roots of x2 – 12x + 35 Step 4 Find the zeros. The zeros appear to be 5 and 7. This means 5 and 7 are the roots of x2 – 12x + 35.

  22.  (–2, 0)    (–1, –2) (–0.5, –2.25). In Your Notes! Example 2a Find the roots of each quadratic polynomial. x2 + x – 2 y = x2 + x – 2 Step 1 Write the related equation. 0 = x2 + x – 2 Step 2 Write the related function. y = x2 + x – 2 Step 3 Graph the related function. • The axis of symmetry is x = –0.5. • The vertex is (–0.5, –2.25). • Two other points (–1, –2) and • (–2, 0) • Graph the points and reflectthem • across the axis of symmetry.

  23. 0 = x2 + x – 2 Check 0 = x2 + x – 2 0(–2)2 + (–2) – 2 012 + (1) – 2 04 – 2 – 2 01 + 1 – 2   0 0 0 0 In Your Notes! Example 2a Continued Find the roots of each quadratic polynomial. Step 4 Find the zeros. The zeros appear to be –2 and 1. This means –2 and 1 are the roots of x2 + x – 2.

  24. (0, 1)   ( , 4) ( , 0). • The axis of symmetry is x = . • The vertex is ( , 0). • Two other points (0, 1) and • ( , 4) • Graph the points and reflectthem • across the axis of symmetry. In Your Notes! Example 2b Find the roots of each quadratic polynomial. 9x2 – 6x + 1 y = 9x2 – 6x + 1 Step 1 Write the related equation. 0 = 9x2 – 6x + 1  Step 2 Write the related function. y = 9x2 – 6x + 1 Step 3 Graph the related function. 

  25. 0 = 9x2 – 6x + 1 Check 09()2 – 6() + 1 01 – 2 + 1  0 0 In Your Notes! Example 2b Continued Find the roots of each quadratic polynomial. Step 4 Find the zeros. There appears to be one zero at . This means that is the root of 9x2 – 6x + 1.

  26.   (1, 6)  • The axis of symmetry is x = . • The vertex is ( , ). • Two other points (1, 6) and • ( , ) • Graph the points and reflectthem • across the axis of symmetry. In Your Notes! Example 2c Find the roots of each quadratic polynomial. 3x2 – 2x + 5 y = 3x2 – 2x + 5 Step 1 Write the related equation. 0 = 3x2 – 2x + 5 Step 2 Write the related function. y = 3x2 – 2x + 5  Step 3 Graph the related function.

  27. In Your Notes! Example 2c Continued Find the roots of each quadratic polynomial. Step 4 Find the zeros. There appears to be no zeros. This means that there are no real roots of 3x2 – 2x + 5.

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