EXAMPLE 1 Solve an equation with two real

EXAMPLE 2 Solve an equation with one real solutions Solve 25 x 2 –

EXAMPLE 2 Solve an equation with one real solutions CHECK Graph y = –

EXAMPLE 3 Solve an equation with imaginary solutions Solve –x 2 + 4 x

EXAMPLE 3 Solve an equation with imaginary solutions CHECK Graph y = 2 x

GUIDED PRACTICE for Examples 1, 2, and 3 Use the quadratic formula to solve

GUIDED PRACTICE for Examples 1, 2, and 3 ANSWER The solutions are x =

GUIDED PRACTICE for Examples 1, 2, and 3 ANSWER The solution is 3 =

GUIDED PRACTICE for Examples 1, 2, and 3 x = – 5 + i

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EXAMPLE 1 Solve an equation with two real solutions Solve x 2 + 3 x = 2 x 2 + 3 x – 2 = 0 x =– b + ANSWER Write original equation. Write in standard form. b 2 – 4 ac 2 a x = – 3 + 32 – 4(1)(– 2) 2(1) Quadratic formula x = – 3 + 17 2 Simplify. The solutions are x = – 3 + 17 2 x = – 3 – 17 – 3. 56. 2 a = 1, b = 3, c = – 2 0. 56 and

EXAMPLE 1 Solve an equation with two real solutions CHECK Graph y = x 2 + 3 x – 2 and note that the x-intercepts are about 0. 56 and about – 3. 56.

EXAMPLE 2 Solve an equation with one real solutions Solve 25 x 2 – 18 x = 12 x – 9. 25 x 2 – 30 x + 9 = 0. x = 30 + Write in standard form. (– 30)2– 4(25)(9) a = 25, b = – 30, c = 9 2(25) 30 + 0 x = 53 ANSWER The solution is 3 5 Write original equation. Simplify.

EXAMPLE 2 Solve an equation with one real solutions CHECK Graph y = – 5 x 2 – 30 x + 9 and note that the only x -intercept is 0. 6 = 3. 5

EXAMPLE 3 Solve an equation with imaginary solutions Solve –x 2 + 4 x = 5 –x 2 + 4 x – 5 = 0. x = – 4+ Write original equation. Write in standard form. 42– 4(– 1)(– 5) a = – 1, b = 4, c = – 5 2(– 1) – 4+ – 4 x= – 2 – 4+ 2 i x= – 2 x=2+i ANSWER The solution is 2 + i and 2 – i. Simplify. Rewrite using the imaginary unit i. Simplify.

EXAMPLE 3 Solve an equation with imaginary solutions CHECK Graph y = 2 x 2 + 4 x – 5. There are no xintercepts. So, the original equation has no real solutions. The algebraic check for the imaginary solution 2 + i is shown. ? 2 –(2 + i) + 4(2 + i) = 5 ? – 3 – 4 i + 8 + 4 i = 5 5=5

GUIDED PRACTICE for Examples 1, 2, and 3 Use the quadratic formula to solve the equation. 1. x 2 = 6 x – 4 SOLUTION x 2 = 6 x – 4 x 2 – 6 x + 4 = 0 Write original equation. Write in standard form. x =– b + b 2 – 4 ac Quadratic formula 2 a x = – (– 6) + (– 6)2 – 4(1)(4) a = 1, b = – 6, c = 4 2(1) x = + 3 + 20 2 Simplify.

GUIDED PRACTICE for Examples 1, 2, and 3 ANSWER The solutions are x = 3 + x= 20 3– 2 =3– 5 20 2 =3+ 5 and

GUIDED PRACTICE for Examples 1, 2, and 3 Use the quadratic formula to solve the equation. 2. 4 x 2 – 10 x = 2 x – 9 SOLUTION 4 x 2 – 10 x = 2 x – 9 4 x 2 – 12 x + 9 = 0 Write original equation. Write in standard form. x =– b + Quadratic formula b 2 – 4 ac 2 a x = – (– 12) + (– 12)2 – 4(4)(9) a = 4, b = – 12, c = 9 2(4) x = 12 + 0 8 Simplify.

GUIDED PRACTICE for Examples 1, 2, and 3 ANSWER The solution is 3 = 1 1. 2 2

GUIDED PRACTICE for Examples 1, 2, and 3 Use the quadratic formula to solve the equation. 3. 7 x – 5 x 2 – 4 = 2 x + 3 SOLUTION 7 x – 5 x 2 – 4 = 2 x + 3 Write original equation. – 5 x 2 + 5 x – 7 = 0 Write in standard form. x =– b + b 2 – 4 ac 2 a x = – (5) + (5)2 – 4(– 5)(– 7) 2(– 5) Quadratic formula x = – 5 + – 115 – 10 Simplify. a = – 5, b = 5, c = – 7

GUIDED PRACTICE for Examples 1, 2, and 3 x = – 5 + i 115 – 10 Rewrite using the imaginary unit i. x = 5 + i 115 10 Simplify. ANSWER The solutions are 5 + i 115 and 10 5 – i 115. 10