1. The Quadratic Formula Objective: To Solve Quadratic Equations using the Quadratic Formula, to write a polynomial function in standard form given its zeros

2. The Quadratic Formula The Quadratic Formula can be used to find the zeros of a quadratic function or the roots (solutions) of a quadratic equation Remember that a quadratic equation has the form ax 2 + bx + c = 0 The Quadratic Formula is:

3. The Quadratic Formula Solve:2x 2 – 3x + 1 = 0 As this is already in standard form, we can see that: a = 2b = -3c = 0 The solution set is {½, 1}

4. The Quadratic Formula Solve:3x 2 + 4x – 2 = 0 As this is already in standard form, we can see that: a = 3b = 4c = -2 The solution set is

5. Finding the Equation of a Quadratic Function Find the zeros of the function:f(x) = x 2 – 3x – 4 x 2 – 3x – 4 = 0 (x – 4)(x + 1) = 0 x – 4 = 0or x + 1 = 0 x = 4x = -1 If we know the zeros of a function, we can work backwards to determine the equation of the function. Step 1:Use the zeros to determine the factors Step 2:If given a value of “a”, write it in front of the factors Step 3:Multiply to get the function into standard form **NOTE: THIS METHOD WORKS ONLY WHEN THE ZEROS ARE INTEGERS**

6. Finding the Equation of a Quadratic Function Find the function with zeros -2 and 5 Given the zeros, we know that x = -2 and x = 5 Working backward, we can find the factors (x + 2) and (x – 5) The function (in factored form) is f(x) = (x + 2)(x – 5) There is no given value of “a,” so the function in standard form is f(x) = x 2 – 3x – 10 Find the function with zeros 1 and 6 and a = 2 x = 1 and x = 6 Factors (x – 1) and (x – 6) f(x) = 2(x – 1)(x – 6) f(x) = 2(x 2 – 7x + 6) f(x) = 2x 2 – 14x + 12

7. Finding the Equation of a Quadratic Function Find the function with zero 4 and a = -3 Because there is only one zero, and we know that a quadratic function must have two zeros, 4 must be a double zero x = 4andx = 4 Factors: (x – 4) and (x – 4) f(x) = -3(x – 4) 2 f(x) = -3(x 2 – 8x + 16) f(x) = -3x + 24x – 48

8. Finding the Equation of a Quadratic Function Find the function with zeros -2 and 6 and y-intercept -24 x = -2 and x = 6 Factors:(x + 2) and (x – 6) f(x) = a(x + 2)(x – 6) To find the value of a, we can substitute the y-intercept coordinates (0, -24) -24 = a(0 + 2)(0 – 6) -24 = a(2)(-6) -24 = -12a a = 2 f(x) = 2(x + 2)(x – 6) f(x) = 2(x 2 – 4x – 12) f(x) = 2x 2 – 8x – 24

9. Finding the Equation of a Quadratic Function Find the function with zeros 3/2 and -1 and a = -4 As the zeros ARE NOT integers, we have to look at the “a” value at the end of the problem x = 3/2and x = -1 2x = 3and x = -1 Factors:(2x – 3) and (x + 1) f(x) = (2x – 3)(x + 1) f(x) = 2x 2 – x – 3 The “a” value once the factors have been multiplied is 2, when it should be -4. We need to adjust the function to correct the “a” value. We do this by multiplying by -2. f(x) = -2(2x 2 – x – 3) f(x) = -4x 2 + 2x + 6

10. Finding the Equation of a Quadratic Function Find the function with zeros -1 and -2 and one point at (-3, -4) x = -1 and x = -2 Factors:(x + 1) and (x + 2) f(x) = a(x + 1)(x + 2) To find the value of a, we can substitute the point coordinates (-3, -4) -4 = a(-3 + 1)(-3 + 2) -4 = a(-2)(-1) -4 = 2a a = -2 f(x) = -2(x + 1)(x + 2) f(x) = -2(x 2 + 3x + 2) f(x) = -2x 2 – 6x – 4