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Given that a parabola y = ax² + bx + c which intersects x-axis at (-2/3,0) and (...

Jul 28, 2024

Given that a parabola y = ax² + bx + c which intersects x-axis at (-2/3,0) and (2,0), intersects y-axis at (0,-2). Find the value of a, b and c.

Solution by Steps

step 1

Given the parabola equation

y = ax^2 + bx + c

, we know it intersects the x-axis at

(-\frac{2}{3}, 0)

and

(2, 0)

. Therefore, we can write the equation as

y = a(x + \frac{2}{3})(x - 2)

step 2

Expand the equation

y = a(x + \frac{2}{3})(x - 2)

to find the coefficients

a

b

, and

c

step 3

y = a(x^2 - 2x + \frac{2}{3}x - \frac{4}{3}) = a(x^2 - \frac{4}{3}x - \frac{4}{3})

step 4

Given that the parabola intersects the y-axis at

(0, -2)

, substitute

x = 0

and

y = -2

into the equation to find

c

step 5

-2 = a(0^2 - \frac{4}{3}(0) - \frac{4}{3}) = -\frac{4}{3}a

step 6

Solve for

a

a = \frac{3}{2}

step 7

Substitute

a = \frac{3}{2}

back into the expanded equation to find

b

and

c

step 8

y = \frac{3}{2}(x^2 - \frac{4}{3}x - \frac{4}{3}) = \frac{3}{2}x^2 - 2x - 2

step 9

Therefore,

a = \frac{3}{2}

b = -2

, and

c = -2

Answer

a = \frac{3}{2}

b = -2

c = -2

Key Concept

Finding the coefficients of a quadratic equation given its roots and a point on the curve

Explanation

By using the given roots and the y-intercept, we can expand the quadratic equation and solve for the coefficients.