CHAPTER EIGHT: SOLVING QUADRATIC EQUATIONS Review April 9 Test April 17 The most important equations at this level of mathematics are quadratic equations. They can be solved using a graph, a perfect square, factors, or a formula called the quadratic formula. 8-A Solving by Graphing Monday 3/12 ➊ Estimate the solutions of an equation by graphing. 8-B Radicals Tuesday 3/13 integer rational irrational radical rationalize the denominator ➊ Identify integers and rational numbers. ➋ Add and subtract radical expressions. ➌ Multiply radical expressions. ➍ Simplify a square root. 8-C Complex Numbers Monday 3/19 i real number imaginary number complex number standard form complex plane complex conjugate ➊ Evaluate the square root of a negative number. ➋ Add and subtract complex numbers. ➌ Multiply complex numbers. ➍ Plot numbers on the complex plane. 8-D Solving with Square Roots Thursday 3/22 complete the square ➊ Solve a quadratic equation by isolating a square. ➋ Solve a quadratic equation by completing the square. 8-E The Quadratic Formula Tuesday 3/27 quadratic formula discriminant ➊ Solve a quadratic equation by using the quadratic formula. ➋ Use the discriminant to determine the number of real solutions of a quadratic equation.
8-A Solving by Graphing Approximate real solutions to an equation can be found by setting the equation equal to zero and finding the x-intercepts. ➊ Estimate the solutions of an equation by graphing. 1. Set the equation equal to zero. 2. Use a grahping calculator or graphing software to graph the eqaution, using y = instead of 0 =. 3. The solutions are the x-values of the x-intercepts (not the x-intercepts themselves). ➊ Solve 0.4x 2 + 3x = -2. 1. 0.4x 2 + 3x + 2 = 0 3. x -6.8, x -0.7 10 5 10-5 0 5 10-5 -10
8-B Radicals An INTEGER is a number that does not require any special symbols other than possibly a negative sign. A RATIONAL Number is an integer divided by an integer. A number that is not rational is IRRATIONAL. ➊ Identify integers and rational numbers. 1. Numbers that can be written without a fraction, decimal, root sign, π, etc., are integers. 2. Numbers that can be written as one integer divided by another are rational. This includes all decimals except those that go on forever without a pattern like π or 2. ➊ Label the following numbers as integers, rational, or irrational. a) -9 b) 0 c) ¾ d) 1.3 e) -23.0996 f) 15 integer integer rational rational rational irrational 1.3 is rational because it is 13 10. -23.0996 is rational because it is -230,996 10,000. An expression under a root sign is a RADICAND. Together with the root sign it is called a RADICAL. Radical expressions can be simplified by combining like terms. Any values multiplied by the same radical, such as 5 3 and -9 3, are like terms. ➋ Add and subtract radical expressions. 1. Distribute any coefficient, including negatives. 2. Combine like terms. ➋ (3 + 5 7) 2(8 2 + 3 2 11 7) 1. 3 + 5 7 16 2 6 2 + 22 7 2. 3 + (5 7 + 22 7) + (-16 2 6 2) 3 + 27 7 22 2 For positive numbers a and b, a b = ab. ➌ Multiply radical expressions. 1. Distribute. 2. Multiply the radicands together. ➌ a) 6 3 b) 6(-3 + 3 5 11) 1. -3 6 + 6 3 6 5 11 2. 18-3 6 + 18 5 66 ➍ Simplify a square root. 1. Factor the radical into two factors, one of which is a square. 2. Take the square root of the square factor, and leave the other factor in the radical. 3. Repeat these steps if there is another square factor in the radical. ➍ 800 1. 100 8 2. 10 8 3. 10 4 2 = 10 2 2 = 20 2
8-C Complex Numbers A REAL Number is a number on the real number line. An IMAGINARY Number is a number that is not on the number line because it involves the square root of a negative. The square root of negative 1 is defined as i: i = -1. A COMPLEX Number is a number that can be written in the form a + bi, where a and b are real numbers and i = -1. Complex numbers written in the form a + bi are in STANDARD Form. Since a or b can be zero, all numbers are complex. However, the term complex number is typically used in the context of imaginary numbers. Imaginary numbers cannot be plotted on the real number line, but they can be plotted on the COMPLEX PLANE. In this plane, the number a + bi is represented by the point (a, b). ➊ Plot numbers on the complex plane. 1. Write the number as a + bi. For real numbers, a = 0. 2. Plot the point (a, b). ➊ Plot the following numbers on the complex plane. a) 2 + 5i b) 2 5i c) 2 d) 5i e) 8 + 11i 2 1. 2 + 5i 2 5i 2 + 0i 0 + 5i 4 + 11 2 i (2, 5) (2, -5) (2, 0) (0, 5) (4, 5.5) The real parts of complex numbers are like terms, and the imaginary parts of complex numbers are like terms. ➋ Add and subtract complex numbers. 1. Distribute any coefficient, including negatives. 2. Find the total of the real parts, and find the total of the imaginary parts. ➋ (3 + 5i) 4(8 + 3i) 1. 3 + 5i 32 12i 2. -29 7i The square root of a negative number -x can be written -x = x -1 = x i. If x is not a whole number, the radical part is written last, after i. ➌ Evaluate the square root of a negative number. 1. Split the number up into the a positive number times -1. 2. Take the square root of the positive number normally (see 8-B), and the square root of -1, which is i. ➌ Evaluate. a) -25 b) -75 25-1 = 5i 25-1 3 = 5i 3
Arithmetically, i functions the same as a variable x, except that i 2 = -1. ➍ Multiply complex numbers. 1. Multiply normally, treating i as a variable. 2. Change every instance of i 2 to -1. ➍ Multiply. a) 3i 4i b) (2 + 3i) (5 6i) c) (5 + 6i)(5 6i) 1. 12i 2 10 + 15i 18i 2 25 30i + 30i 36i 2 2. -12(-1) 10 + 15i 18(-1) 25 30i + 30i 36(-1) -12 10 + 15i + 18 25 + 36 28 + 15i 61 The COMPLEX CONJUGATE of a number a + bi is a bi. A number multiplied by its complex conjugate is a real number, as in the last example above.
8-D Solving with Square Roots If one side of an equation is a square and the other side is a constant, the equation can be solved by taking the square root of each side. ➊ Solve a quadratic equation by isolating a square. 1. Rewrite the equation so that there is a squared polynomial alone on one side and a constant alone on the other. This may involve adding a value to each side and/or dividing each side by a value. 2. Take the square root of each side. 3. Solve the + equation. 4. Solve the - equation. ➊ 8(2x + 3) 2 80 = 120 1. 8(2x + 3) 2 = 200 (2x + 3) 2 = 25 2. 2x + 3 = ±5 3. 2x + 3 = 5 x = 1 4. 2x + 3 = -5 x = -4 If there isn t already a squared polynomial in a quadratic equation, you can make one by COMPLETING THE SQUARE. ➋ Solve a quadratic equation by completing the square. 1. Subtract c from each side. 2. Divide each term by a. 3. Complete the square by adding ( b 2) 2 to each side. 4. Rewrite the square side as (x + b 2) 2. 5. Square root each side. 6. Solve the + equation. 7. Solve the - equation. ➋ 2x 2 + 20x + 34 = 16 1. 2x 2 + 20x = -18 2. x 2 + 10x = -9 3. x 2 + 10x + 25 = -9 + 25 4. (x + 5) 2 = 16 5. x + 5 = ±4 6. x + 5 = 4 x = -1 x + 5 = -4 x = -9
8-E The Quadratic Formula The QUADRATIC FORMULA is x = -b ± b2 4ac, where a, b, and c are the coefficients of the quadratic equation ax 2a 2 + bx + c = 0. Any quadratic equation can be solved by the quadratic formula. ➍ Solve a quadratic equation with the quadratic formula. 1. Put the equation in standard form. 2. Identify a, b, and c. 3. Plug a, b, and c into the quadratic formula. 4. Simplify. ➍ 12x 2 + 17x = 7 1. 12x 2 + 17x 7 = 0 2. a = 12, b = 17, c = -7 3. x = -17 ± 172 4(12)(-7) 2(12) -17 ± 25 4. x = 24 = 1 3 or -7 4 THE DISCRIMINANT of a Quadratic Polynomial in standard form is b 2 4ac, which is the radicand of the quadratic formula. Since the positive and negative square roots of this will be found in order to find the roots of the polynomial, the two roots of the polynomial are imaginary if b 2 4ac is negative. If b 2 4ac is positive, then there are two distinct real roots. If b 2 4ac is zero, then the two roots are real and identical, making for only one distinct real root. Likewise, the discriminant determines the number of real solutions, zeros, or x-intercepts of a quadratic. ➋ Use the discriminant to determine the number of real solutions of a quadratic equation. 1. Put the quadratic in standard form ax 2 + bx + c = 0. 2. Evaluate the discriminant b 2 4ac. 3. If the discriminant is positive, there are two real solutions. If the discriminant is zero, there is one real solution. If the discriminant is negative, there are two imaginary solutions. ➋ 2x 2 = 4x 5 1. 2x 2 4x + 5 = 0 2. The discriminant is (-4) 2 4(2)(5) = -24 3. -24 is negative, so there are two imaginary solutions.