Question 1 :
The graph of a parabola in the xy-plane has x-intercepts 3/5 and -1/2. Which of the following could be the equation of the parabola?
A) y = (5x - 1)((2x + 3)
B) y = (5x + 1)(2x - 3)
C) y = (5x - 3)(2x + 1)
D) y = (5x + 3)(2x - 1)
Answer :
To find x-intercepts of any curve, we have to plugin y = 0 into its equation and find the values of x.
In each of the answer choices, plugin y = 0 and find x-intercepts.
A) y = (5x - 1)((2x + 3) :
(5x - 1)((2x + 3) = 0
5x - 1 = 0 or 2x + 3 = 0
x = 1/5 or x = -3/2
x-intercepts are 1/5 and -3/2.
B) y = (5x + 1)((2x - 3) :
(5x + 1)((2x - 3) = 0
5x + 1 = 0 or 2x - 3 = 0
x = -1/5 or x = 3/2
x-intercepts are -1/5 and 3/2.
C) y = (5x - 3)(2x + 1) :
(5x - 3)((2x + 1) = 0
5x - 3 = 0 or 2x + 1 = 0
x = 3/5 or x = -1/2
x-intercepts are 3/5 and -1/2.
D) y = (5x + 3)(2x - 1) :
(5x + 3)((2x - 1) = 0
5x + 3 = 0 or 2x - 1 = 0
x = -3/5 or x = 1/2
x-intercepts are -3/5 and 1/2.
The correct answer choice is (C).
Question 2 :
If k^{2} + 4k = 45 and k > 0, what is the value of (k + 2)?
Answer :
k^{2} + 4k = 45
Subtract 45 from both sides.
k^{2} + 4k - 45 = 0
(k + 9)(k - 5) = 0
k + 9 = 0 or k - 5 = 0
k = -9 or k = 5
Since k > 0, k = 5.
k + 2 = 5 + 2
k + 2 = 7
Question 3 :
The function f is defined by f(x) = x^{2} + bx + c, where b and c are constants. If the graph of f has x-intercepts at -5 and 3, which of the following correctly gives the values of b and c?
A) b = -5 and c = 3
B) b = -3 and c = 5
C) b = -2 and c = -15
D) b = 2 and c = -15
Answer :
Given : x-intercepts are -5 and 3.
That is, when f(x) = 0, x = -5 or x = 3.
(-5)^{2} + b(-5) + c = 0 25 - 5b + c = 0 -5b + c = -25 ----(1) |
3^{2} + b(3) + c = 0 9 + 3b + c = 0 3b + c = -9 ----(2) |
(2) - (1) :
(3b + c) - (-5b + c) = -9 - (-25)
3b + c + 5b - c = -9 + 25
8b = 24
b = 3
Substitute b = 3 in (2).
3(3) + c = -9
9 + c = -9
c = -18
Question 4 :
What is the sum of the solutions to 2x^{2} - 6x + 2 = 0?
A) -3
B) -1
C) 1
D) 3
Answer :
2x^{2} - 6x + 2 = 0
In the quadratic equation above, the coefficients of x2, x and constant are all divisible by 2. So, divide both sides of the equation by. to make the values smaller.
x^{2} - 3x + 1 = 0
Comparing x^{2} - 3x + 1 = 0 and ax^{2} + bx + c = 0,
a = 1, b = -3 and c = 1
Formula to find the sum of the solutions of a quadratic equation in standard form ax^{2} + bx + c = 0 :
= -b/a
Substitute a = 1 and b = -3.
= -(-3)/1
= 3
The correct answer choice is (D).
Question 5 :
x^{2} - 5x + c = 0
In the quadratic equation, c is a constant. If the equation has two solutions for x, one of which is -3, what is the value of the other solution?
Answer :
Given : For the given quadratic equation, one of the solutions is -3.
Let k be the other solution.
Comparing x^{2} - 5x + c = 0 and ax^{2} + bx + c = 0,
a = 1, b = -5 and c = c
sum of the solutions = -b/a
-3 + k = -(-5)/1
-3 + k = 5
k = 8
The other solution is 8.
Question 6 :
y = x^{2} - 2x - 3
A parabola in the xy-plane is given by the equation above. Which of the following equivalent forms of the equation displays the coordinates of the vertex of parabola as constants or coefficients?
A) y = (x - 1)^{2} - 4
B) y = (x - 1)^{2} - 2
C) y = (x - 3)(x + 1)
D) y + 3 = x(x + 2)
Answer :
The vertex form equation of a parabola y = a(x - h)^{2} + k, displays the coordinates of the vertex (h, k) as constants or coefficients.
Write the given equation of the parabola in vertex form using completing the square method.
y = x^{2} - 2x - 3
y = x^{2} - 2(x)(1) - 3
y = x^{2} - 2(x)(1) + 1^{2} - 1^{2} - 3
Using the algebraic identity, (a + b)^{2} = a^{2} + 2ab + b^{2},
y = (x - 1)^{2} - 1^{2} - 3
y = (x - 1)^{2} - 1 - 3
y = (x - 1)^{2} - 4
The correct answer choice is open (A).
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