To create a quadratic function from zeroes we have to follow the formula given below.
x^{2} - (sum of the roots)x + product of the roots = 0
Other names of zeroes :
Zeroes, x-intercepts, roots, solutions.
Problem 1 :
Write a quadratic equation in standard form with solutions, x = -3 and x = 4. Use integers for a, b and c.
Solution:
Sum of the roots :
= -3 + 4
= 1
Product of the roots :
= -3 × 4
= -12
Formation of quadratic equation:
x^{2} - (sum of the roots)x + product of the roots = 0
x^{2} - x - 12 = 0
Problem 2 :
Write a quadratic equation in standard form with solutions, x = 2/3 and x = -5. Use integers for a, b and c.
Solution:
Sum of the roots :
= (2/3) + (-5)
= (2/3) - 5
= (2 - 15)/3
= -13/3
Product of the roots :
= (2/3) × (-5)
= -10/3
Formation of quadratic equation:
x^{2} - (sum of the roots)x + product of the roots = 0
x^{2} - (-13/3)x + (-10/3) = 0
x^{2} + (13/3)x - (10/3) = 0
Multiply each side by 3.
3x^{2} + 13x - 10 = 0
Problem 3 :
Write an equation of the parabola in intercept form y = a(x - p)(x - q) that has x- intercepts of 9 and 1 and passes through (0, -18).
A) y = -1/2(x - 9)(x - 1) B) y = -1/2(x + 9)(x + 1)
C) y = -2(x - 9)(x - 1) D) y = -2(x + 9)(x + 1)
Solution:
y = a(x - p)(x - q)
Using the given x-intercepts, we can write
y = a(x - 9)(x - 1)
It is passes through the point (x, y) = (0, -18).
-18 = a(0 - 9)(0 - 1)
-18 = a(-9)(-1)
-18 = 9a
a = -2
So, the equation of the parabola in intercept form is
y = -2(x - 9)(x - 1)
So, option (C) is correct.
Problem 4 :
Write an equation of the parabola in intercept form y = a(x - p)(x - q) that has x- intercepts of 12 and -6 and passes through (14, 4).
A) y = 1/10(x - 12)(x + 6) B) y = 1/10(x + 12)(x - 6)
C) y = 10(x - 12)(x + 6) D) y = 10(x + 12)(x - 6)
Solution:
y = a(x - p)(x - q)
Using the given x-intercepts, we can write
y = a(x - 12)(x + 6)
It is passes through the point (x, y) = (14, 4).
4 = a(14 - 12)(14 + 6)
4 = a(2)(20)
4 = 40a
a = 1/10
So, the equation of the parabola in intercept form is
y = 1/10(x - 12)(x + 6)
So, option (A) is correct.
Problem 5 :
Determine the equation of a quadratic function given zeros x = 4 and point (3, 2).
Solution:
y = a(x - p)(x - q)
Given x- intercept = 4
y = a(x - 4)(x - 4)
To find a, substitute the point (x, y) = (3, 2).
2 = a(3 - 4)(3 - 4)
2 = a(-1)(-1)
a = 2
y = 2(x - 4)(x - 4)
Problem 6 :
Use the intercepts and a point on the graph below to write the equation of the function.
Solution:
y = a(x - p)(x - q)
x- intercept = (-6, 0) and (6, 0)
y = a(x + 6)(x - 6)
To find a, substitute the point (x, y) = (-4, -20).
-20 = a(-4 + 6)(-4 - 6)
-20 = a(2)(-10)
-20= -20a
a = 1
y = (x + 6)(x - 6)
Problem 7 :
Use the intercepts and a point on the graph below to write the equation of the function.
Solution:
y = a(x - p)(x - q)
x- intercept = (-6, 0) and (1, 0)
y = a(x + 6)(x - 1)
To find a, substitute the point (x, y) = (-3, -36).
-36 = a(-3 + 6)(-3 - 1)
-36 = a(3)(-4)
-36= -12a
a = 3
y = 3(x + 6)(x - 1)
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May 21, 24 08:51 AM
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