Topic 5 1 Radical Equations A radical equation is an equation with at least one radical expression. There are four types we will cover: x 35 3 4x x 1x 7 3 3 3 x 5 x 1 To solve a radical equation, you must take both sides of an equation to a power. Beware; solving a radical equation can produce false answers. False solutions are called extraneous. Thus, you must always check any solutions you find with a radical equation. x 3 5 x x 53
3 4x 3 5 x 1 10
4 5x 1 3 6 x 4
x 1x 7
x 5x 3
x 3 x
3x 5 x 1 3 3 3 x 5 x 1
x x 53
x 1 x 4
Topic 5 Imaginary Numbers Simplify. Definition: Let i be defined as the imaginary base number such that i 1. Then i 1. 1 3 Definition: An imaginary numbers is any number that can be simplified into the form bi, where b is a real number coefficient.
Multiply and/or divide. 8 4 6 Powers of i i i 1 i 1 3 i i i1ii 4 i i i 111 An example of cyclic behavior: 5 4 i i i1ii 6 4 i i i 111 7 4 3 i i i 1ii 8 4 4 i i i 11 1
Simplify. Simplify. 311 i 536 i 4i 3
Topic 5 3 Complex Numbers Add and/or subtract. 63i 5i 63i 5i
Add and/or subtract. 3i73i5i Multiply. 63i5 i
Multiply. i 7 4i Multiply. 3 5i
Multiply. 6i6i Divide. 5 8i i
Divide. 5 i 6 3i Divide. 7 3i 4i
Topic 5 4 Property of Square Roots Property of Square Roots If x a, then x a. x 18 If the square of an unknown equals a number, then the unknown is equal to the positive square root of the number and the negative square root of the number. x 5
x 1 0 x 3 49
x 1 5 x 5
x 4 4 x 3 70
Topic 5 5 Completing the Square The expression completing the square refers to taking a quadratic expression and adding an appropriate constant so that the expression is a perfect square. Recall: A quadratic expression is called a perfect square if it is factorable into a squared binomial expression. Determine what should be added to make each expression a perfect square. Then factor. x² 10x x² + x x² 4x x² + 14x
Process for completing the square given x² + bx 1. Multiply 1 to b. (Same as dividing b by.) Determine what should be added to make each expression a perfect square. Then factor. x² + 3x. Square the number found in step 1. 3. Add the number from step to x² + bx. The expression should not be factorable into (x + 1 b)² x² + 3 4 x
Strategy for solving an equation by completing the square: 1. If not given as such, write the equation in the form ax² + bx + c = 0. x 6x 70. If necessary, divide a through everything. 3. Add or subtract a number to move c to the other side. 4. Complete the square on one side and balance the equation on the other side. 5. Factor and simplify. 6. Use the Property of Square Roots to find solutions to the equation.
x 10x 3 0 x 8x 40
x 5x 10 3x x 90
4x 8x 1 0 x 10x 33 0
Topic 5 6 Quadratic Formula For a quadratic equation in the form ax² + bx + c = 0, the b b 4ac solutions can be found by x. a x 6x 50 x 6x 70
4x x 60 4x 9x 0
9x 8 5x 3 x x 5 0 8
x x 1 3 4 The discriminant D of a quadratic function is equal to the radicand of the quadratic formula; that is, D = b 4ac The nature of the solutions to a quadratic function can be determined by the discriminant. 1. If D > 0, the quadratic function has two real solutions.. If D = 0, the quadratic function has exactly one real solution. 3. If D < 0, the quadratic function has two nonreal solutions.