Algebra 1B Chapter 9 Solving Quadratic Equations By Graphing

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

Every quadratic function has a related quadratic equation. The standard form of a quadratic equation is ax 2 + bx + c = 0, where a, b, and c are real numbers and a ≠ 0. y = ax 2 + bx + c 0 = ax 2 + bx + c When writing a quadratic function as its related quadratic equation, you replace y with 0.

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.

Additional Example 1A: Solving Quadratic Equations by Graphing Solve the equation by graphing the related function. 2x 2 – 18 = 0 Step 1 Write the related function. 2x 2 – 18 = y, or y = 2x 2 + 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 reflect them across the axis of symmetry. (3, 0) ● x = 0 (2, –10) ● (0, –18) ● ● ●

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

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

Additional Example 1C: Solving Quadratic Equations by Graphing 2x 2 + 4x = –3 Step 1 Write the related function. y = 2x 2 + 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 reflect them across the axis of symmetry.  (–1, 1)     (0, 3) (1, 9) (–2, 3) (–3, 9) Solve the equation by graphing the related function.

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

In Your Notes! Example 1a Solve the equation by graphing the related function. x 2 – 8x – 16 = 2x 2 Step 1 Write the related function. y = x 2 + 8x + 16 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 reflect them across the axis of symmetry. x = –4 (–4, 0) ● (–3, 1) ● (–2, 4) ● ● ●

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

Solve the equation by graphing the related function. 6x + 10 = –x 2 Step 1 Write the related function. y = x 2 + 6x + 10 In Your Notes! Example 1b 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 reflect them across the axis of symmetry. x = –3 (–3, 1) ● (–2, 2) ● (–1, 5) ● ● ●

Solve the equation by graphing the related function. x 2 + 6x + 10 = 0 In Your Notes! Example 1b Continued The equation has no real-number solutions. Step 3 Find the zeros. The function appears to have no zeros

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

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

Additional Example 2A: Finding Roots of Quadratic Polynomials Find the roots of x 2 + 4x + 3 Step 1 Write the related equation. 0 = x 2 + 4x + 3 Step 2 Write the related function. Step 3 Graph the related function. y = x 2 + 4x + 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 reflect them across the axis of symmetry. y = x 2 + 4x + 3      (–2, –1) (–3, 0) (–4, 3)

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 x 2 + 4x + 3. Check 0 = x 2 + 4x (–3) 2 + 4(–3) – = x 2 + 4x (–1) 2 + 4(–1) – 4 + 3

Additional Example 2B: Finding Roots of Quadratic Polynomials Find the roots of x 2 + x – 20 Step 1 Write the related equation. 0 = x 2 + x – 20 Step 2 Write the related function. Step 3 Graph the related function. y = x 2 + 4x – 20 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 reflect them across the axis of symmetry. y = x 2 + 4x – 20 (–0.5, –20.25).  (1, –18)  (2, –15)   

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

Additional Example 2C: Finding Roots of Quadratic Polynomials Find the roots of x 2 – 12x + 35 Step 1 Write the related equation. 0 = x 2 – 12x + 35 y = x 2 – 12x + 35 Step 2 Write the related function. 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 reflect them across the axis of symmetry. (6, –1).  (4, 3)  (5, 0)   

Additional Example 2C Continued Find the roots of x 2 – 12x + 35 Step 4 Find the zeros. The zeros appear to be 5 and 7. This means 5 and 7 are the roots of x 2 – 12x Check 0 = x 2 – 12x – 12(5) – = x 2 – 12x – 12(7) –

In Your Notes! Example 2a Find the roots of each quadratic polynomial. x 2 + x – 2 Step 1 Write the related equation. 0 = x 2 + x – 2 Step 2 Write the related function. Step 3 Graph the related function. y = x 2 + x – 2 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 reflect them across the axis of symmetry.  (–0.5, –2.25). (–1, –2)  (–2, 0)    y = x 2 + x – 2

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 x 2 + x – 2. Check 0 = x 2 + x – (–2) 2 + (–2) – – 2 – 2 0 = x 2 + x – (1) – – 2 In Your Notes! Example 2a Continued

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

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

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

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