Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Lagrange multiplier wikipedia , lookup
Mean field particle methods wikipedia , lookup
Finite element method wikipedia , lookup
Linear least squares (mathematics) wikipedia , lookup
False position method wikipedia , lookup
Dynamic substructuring wikipedia , lookup
Newton's method wikipedia , lookup
Root-finding algorithm wikipedia , lookup
Cubic function wikipedia , lookup
System of linear equations wikipedia , lookup
System of polynomial equations wikipedia , lookup
Math 20-1 Quadratic Functions and Equations Name: ____________________ Solving Quadratic Equations Specific Outcome R&F 5: Solve problems that involve quadratic equations. Questions all Total 47 Solving a quadratic equation means finding what value of the variable makes both sides of the equation equal. Solutions have the form x ______ . A quadratic equation can have two, one, or no solutions. There are three ways to solve a quadratic equation algebraically: 1. Factoring 2. Completing the Square 3. Quadratic Formula For all methods of solving, you must first move all the terms to one side so that the equation = 0. Method 1: Solving by Factoring * - GCF - Difference of squares - Trinomial factoring (PSF / Grid) Then set each factor = 0 and solve 1. Find the roots of the following equations by factoring. (2 marks each) a) x 2 6 x 27 b) 4h 2 49 0 c) 1 2 d 3d 4 d) n 3n 4 6 Math 20-1 Quadratic Functions and Equations e) x 2 7 x20 2 f) 9 x 2 7 x 6 5 x 2 2. A field is in the shape of a right triangle. The fence around the perimeter of the field is 40 m in total. The length of the hypotenuse is 17 m. Find the length of the other two sides of the right triangle by writing and solving a quadratic equation by factoring. (3 marks) Math 20-1 Quadratic Functions and Equations Method 2: Solving by Completing the Square - Best if the equation is already in vertex form Don’t forget the square root to get two solutions! 3. Find the roots of the following quadratic equations by completing the square. Write your answers in exact value, if necessary. 1 2 a) x 8 1 0 (2 marks) 4 b) 5 x 2 10 x 2 0 (3 marks) Math 20-1 Quadratic Functions and Equations Method 3: Solving by Quadratic Formula - b b 2 4ac Quadratic formula: x 2a Rounded to the nearest tenth, hundredth, etc: two answers Exact value – simplify the radical (and the fraction, if possible) Square root of a negative means there are no real roots. 4. Find the roots of the following quadratic equations using the quadratic formula. Write your answers as integers or fractions if possible, or round to the nearest hundredth, if necessary. (2 marks each) a) 4 x 2 8 x 3 0 b) 3 x 2 4 x 7 5. Solve the following equations by using the quadratic formula. Write all answers in simplest exact form. (3 marks each) a) 5n 2 7 11n b) 1 x 2 4 x Math 20-1 Quadratic Functions and Equations 6. The surface area of a cone is 300 cm2 and its slant height is 15 cm. Use the surface area formula A r 2 rs to determine the radius of the cone to the nearest hundredth of a centimetre. (3 marks) 7. A rectangle has a length that is 7 ft longer than its width. The numerical value of the area of the rectangle (in ft2) is equal to the numerical value of the perimeter of the rectangle (in ft). Find the dimensions of the rectangle to the nearest hundredth, and its area and perimeter to the nearest tenth. (4 marks) Math 20-1 Quadratic Functions and Equations 8. Find the value of the discriminant for each quadratic equation. Then say how many roots the equation has. (2 marks each) a) 5 x 2 4 x 12 0 b) 36 x 2 12 x 1 0 c) 8 x 2 x 2 3 d) x 2 10 14 x 10 9. For what values of c does 18 x 3x 2 c have no real roots? (2 marks)