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Quadratic Equation passing through two points










































Find coefficients and x-intercepts of a Quadratic Equation

• A quadratic graph:

       1. y = 2x² + bx + c

       2. passes through points A (0,5) and B(-1,0).

• Find

       (i) the value of c

       (ii) the value of b

       (iii) the x-intercepts

Comments for Quadratic Equation passing through two points

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Aug 29, 2012
Quadratic Equation
by: Staff


Answer:


Part I

The standard form of a quadratic function (a parabola) is:

y = ax² + bx + c

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y = 2x² + bx + c

a = 2


(i) the value of “c”


       The quadratic function (a parabola) passes through point A (0,5).

           When x = 0, y = 5

           To solve for the coefficient “c”, substitute 0 for x and 5 for y in the equation given in the problem statement.

               y = 2x² + bx + c

               5 = 2*0² + b*0 + c

               5 = 0 + 0 + c

               5 = c

               c = 5


(ii) the value of “b”

       The quadratic function also passes through point B (-1,0).

           When x = -1, y = 0

           To solve for the coefficient “b”, substitute -1 for x, 0 for y, and 5 for c in the quadratic function.

               y = 2x² + bx + c


               0 = 2*(-1)² + b*(-1) + 5

               0 = 2*1 + b*(-1) + 5

               0 = 2 - b + 5

               0 = 2 + 5 - b

               0 = 7 - b

               0 + b = 7 - b + b

               b = 7 + 0

               b = 7

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Aug 29, 2012
Quadratic Equation
by: Staff


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Part II

a = 2

b = 7

c = 5


y = ax² + bx + c


y = 2x² + 7x + 5

Check the final equation:

If the quadratic function passes through point A (0,5), then y must equal 5 when x = 0

y = 2*0² + 7*0 + 5

y = 0 + 0 + 5

y = 5, OK


If the quadratic function passes through point B (-1,0), then y must equal 0 when x = -1

y = 2*(-1)² + 7*(-1) + 5

y = 2*1 - 7*1 + 5

y = 2 - 7 + 5

y = 0, OK

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(iii) the x-intercepts


       The final form of the quadratic function is: y = 2x² + 7x + 5

       The quadratic equation is: 2x² + 7x + 5 = 0


           the x-intercepts can be determined using the quadratic formula, or by factoring.


           Determining the x-intercepts by factoring:

               2x² + 7x + 5 = 0

               (2x + 5)(x + 1) = 0

               (2x + 5) = 0


               2x + 5 = 0

               2x + 5 - 5 = 0 - 5


               2x + 0 = 0 - 5


               2x = - 5

               2x / 2 = - 5 / 2


               x * (2 / 2) = - 5 / 2


               x * (1) = - 5 / 2


               x = - 5 / 2

               the first x-intercept, x₁ = - 5 / 2

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Aug 29, 2012
Quadratic Equation
by: Staff

---------------------------------------------

Part III

               (2x + 5)(x + 1) = 0


               (x + 1) = 0


               x + 1 = 0

               x + 1 - 1 = 0 - 1


               x + 0 = 0 - 1


               x = - 1

               the second x-intercept, x₂ = - 1



 Graph of Quadratic Function y = 2x² + 7x + 5




summary of final answers

     (i) c = 5

     (ii) b = 7

     (iii) the x-intercepts ∈ {-5/2, -1}






Thanks for writing.

Staff
www.solving-math-problems.com



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