2 Standard form of a quadratic function A quadratic equation is a equation that can be written in the form𝒂 𝒙 𝟐 +𝒃𝒙+𝒄=𝟎Where 𝑎≠0. This form is called the standard form of a quadratic equation
3 Quadratic equation can be solved by a variety of methods GraphingSquared Root propertyFactoringCompleting the squareQuadratic formula
4 Solutions of a Quadratic Equation A quadratic equation can have two, one or no real number solutionsThe solutions of a quadratic equation are the x- intercepts of the related functionThe solutions of a quadratic equation are often calledRoots of the equation orZeros of the function
5 Steps in Solving Quadratic Equations If the equation is in the form (ax+b)2 = c, use the square root property to solve.If not solved in step 1, write the equation in standard form.Try to solve by factoring.If you haven’t solved it yet, use the quadratic formula.
6 are the x-intercepts 2 and -2 Solving Quadratic Equation by graphingGraph the parabola by:yxFinding the equation of the axis of symmetryThe solution of𝑥 2 −4=0are the x-intercepts 2 and -2𝑥= −𝑏 2𝑎 x= 0 2(1) 𝑥=0Finding the vertex(0) 2 −4=− Vertex = (0,-4)Making a table using the x-values around the axis of symmetry2-2x𝒙 𝟐 −𝟒y2(𝟐) 𝟐 −𝟒-2(−𝟐) 𝟐 −𝟒Graphing each point on a coordinate planeThe roots of the equation or solution are the x-intercept or zeros of the related quadratic function
7 Examples of solving by graphing What are the solution of each equation? Use a graph of the related function𝑥 2 −1=0𝑥 2 =0𝑥 2 +1=0There are two solutions1 and -1There is one solutionsThere is no real number solution
8 Solving Quadratic Equation by using the square root Square Root PropertyIf b is a real number and a2 = b, thenYou can solve equations of the form𝒙 𝟐 =𝒌by finding the square root of each side. For example𝑥 2 =81𝑥=± 81𝑥=±9
9 Solving Quadratic Equation by using the square root ExampleSolve:𝐵) 3𝑥 2 −75=0A) (y – 3)2 = 43𝑥 2 =75𝑥 2 = 75 3𝑦−3=±2𝑥 2 = 25𝑦=3±2𝑦=1 𝑜𝑟 𝑦=5𝑥=± 25𝑥=±5
10 Solving Quadratic Equation by using Quadratic formulaThis method will work to solve ALL quadratic equationsFor many equations it takes longer than some of the other methods.
11 Solving Quadratic Equation by using Quadratic formulaA quadratic equation written in standard form, ax2 + bx + c = 0, has the solutions.𝒙= −𝒃± 𝒃 𝟐 −𝟒𝒂𝒄 𝟐𝒂When getting your solution, if the radicand in the quadratic formula is not a perfect square, you can use a calculator to approximate the solution
12 Solve x2 + x – = 0 by the quadratic formula. ExampleSolve x2 + x – = 0 by the quadratic formula.x2 + 8x – 20 = (multiply both sides by 8)a = 1, b = 8, c = 20= − or −8−12 2= −20 2 𝑜𝑟 4 2𝑥=−10 𝑜𝑟 𝑥=2
13 So there is no real solution. ExampleSolve x(x + 6) = 30 by the quadratic formula.(Simplify the polynomial and write it on standard form)x2 + 6x + 30 = 0a = 1, b = 6, c = 30So there is no real solution.Square root can’t be negative.
14 Solve 2x = x2 - 8. x2 – 2x – 8 = 0 Let a = 1, b = -2, c = -8 Example write the equation on standard formx2 – 2x – 8 = Let a = 1, b = -2, c = -8𝑥= 𝑜𝑟𝑥= 2−6 2𝑥= 𝑜𝑟𝑥=−2
15 The DiscriminatBefore you solve a quadratic equation you can determine how many real-number solutions it has by using discriminant.The expression under the radical sign in the formula (b2 – 4ac) is called the discriminant.The discriminant will take on a value that is positive, 0, or negative.The value of the discriminant indicates two distinct real solutions, one real solution, or no real solutions, respectively.
17 ExampleUse the discriminant to determine the number and type of solutions for the following equation.5 – 4x + 12x2 = 0a = 12, b = –4, and c = 5b2 – 4ac = (–4)2 – 4(12)(5)= 16 – 240= –224Because the discriminant is negative, the equation has no-real number solution.
18 Because the discriminant is positive, the equation has two solution. ExampleUse the discriminant to determine the number and type of solutions for the following equation.6 𝑥 2 −5𝑥=76 𝑥 2 −5𝑥−7=0a = 6, b = –5, and c = -7b2 – 4ac = (–6)2 – 4(6)(-7)== 232Because the discriminant is positive, the equation has two solution.