Determine the value of the coefficient \( a \) in the following equation:
\( -x^2+7x-9 \)
Incorrect
Correct Answer:
-1
Question 2
What is the value of the coefficient \( b \) in the equation below?
\( 3x^2+8x-5 \)
Incorrect
Correct Answer:
8
Question 3
What is the value of the coefficient \( c \) in the equation below?
\( 4x^2+9x-2 \)
Incorrect
Correct Answer:
-2
Question 4
\( y=x^2 \)
Incorrect
Correct Answer:
\( a=1,b=0,c=0 \)
Question 5
\( y=x^2+10x \)
Incorrect
Correct Answer:
\( a=1,b=10,c=0 \)
Examples with solutions for The Quadratic Function
Exercise #1
Determine the value of the coefficient a in the following equation:
−x2+7x−9
Video Solution
Step-by-Step Solution
The quadratic equation in the problem is already arranged (meaning all terms are on one side and 0 on the other side), so let's proceed to answer the question asked:
The question asked in the problem - What is the value of the coefficienta in the equation?
Let's recall the definitions of coefficients in solving quadratic equations and the roots formula:
The rule states that the roots of an equation of the form:
ax2+bx+c=0are:
x1,2=2a−b±b2−4ac
That is the coefficient ais the coefficient of the quadratic term (meaning the term with the second power)- x2Let's examine the equation in the problem:
−x2+7x−9=0
Remember that the minus sign before the quadratic term means multiplication by: −1 , therefore- we can write the equation as:
−1⋅x2+7x−9=0
The number that multiplies the x2, is −1 hence we identify that the coefficient of the quadratic term is the number −1,
Therefore the correct answer is A.
Answer
-1
Exercise #2
What is the value of the coefficient b in the equation below?
3x2+8x−5
Video Solution
Step-by-Step Solution
The quadratic equation of the given problem is already arranged (that is, all the terms are found on one side and the 0 on the other side), thus we approach the given problem as follows;
In the problem, the question was asked: what is the value of the coefficientbin the equation?
Let's remember the definitions of coefficients when solving a quadratic equation as well as the formula for the roots:
The rule says that the roots of an equation of the form
ax2+bx+c=0are :
x1,2=2a−b±b2−4ac
That is the coefficientbis the coefficient of the term in the first power -xWe then examine the equation of the given problem:
3x2+8x−5=0That is, the number that multiplies
x is
8Consequently we are able to identify b, which is the coefficient of the term in the first power, as the number8,
Thus the correct answer is option d.
Answer
8
Exercise #3
What is the value of the coefficient c in the equation below?
4x2+9x−2
Video Solution
Step-by-Step Solution
The quadratic equation is given as 4x2+9x−2. This equation is in the standard form of a quadratic equation, which is ax2+bx+c, where a, b, and c are coefficients.
The term 4x2 indicates that the coefficient a=4.
The term 9x indicates that the coefficient b=9.
The constant term −2 indicates that the coefficient c=−2.
From this analysis, we can see that the coefficient c is −2.
Therefore, the value of the coefficient c in the equation is −2.
Answer
-2
Exercise #4
y=x2
Video Solution
Step-by-Step Solution
To solve this problem, let's follow these steps:
Step 1: Recognize the standard form of a quadratic equation.
Step 2: Match the given function to the standard form.
Step 3: Identify each coefficient a, b, and c.
Now, let's work through these steps:
Step 1: The standard form of a quadratic function is y=ax2+bx+c. Our goal is to identify a, b, and c.
Step 2: We are given the function y=x2. This can be aligned with the standard form as y=1⋅x2+0⋅x+0.
Step 3: By comparing the given function y=x2 with the standard form, we can deduce:
- The coefficient of x2 is 1, so a=1.
- The linear term coefficient is missing, which implies b=0.
- There is no constant term, so c=0.
Therefore, the coefficients are a=1,b=0,c=0, corresponding to choice 1.
Answer
a=1,b=0,c=0
Exercise #5
y=x2+10x
Video Solution
Step-by-Step Solution
Here we have a quadratic equation.
A quadratic equation is always constructed like this:
y=ax2+bx+c
Where a, b, and c are generally already known to us, and the X and Y points need to be discovered.
Firstly, it seems that in this formula we do not have the C,
Therefore, we understand it is equal to 0.
c=0
a is the coefficient of X², here it does not have a coefficient, therefore
a=1
b=10
is the number that comes before the X that is not squared.
Answer
a=1,b=10,c=0
Question 1
\( y=x^2-6x+4 \)
Incorrect
Correct Answer:
\( a=1,b=-6,c=4 \)
Question 2
\( y=2x^2-5x+6 \)
Incorrect
Correct Answer:
\( a=2,b=-5,c=6 \)
Question 3
\( y=2x^2-3x-6 \)
Incorrect
Correct Answer:
\( a=2,b=-3,c=-6 \)
Question 4
\( y=-2x^2+3x+10 \)
Incorrect
Correct Answer:
\( a=-2,b=3,c=10 \)
Question 5
\( y=3x^2+4x+5 \)
Incorrect
Correct Answer:
\( a=3,b=4,c=5 \)
Exercise #6
y=x2−6x+4
Video Solution
Step-by-Step Solution
To solve this problem, we'll clearly delineate the given expression and compare it to the standard quadratic form:
Step 1: Recognize the standard form of a quadratic equation as y=ax2+bx+c.
Step 2: Compare the given equation y=x2−6x+4 to the standard form.
Step 3: Identify coefficients:
- The coefficient of x2 is a=1.
- The coefficient of x is b=−6.
- The constant term is c=4.
Therefore, the coefficients for the quadratic function y=x2−6x+4 are a=1, b=−6, and c=4.
Among the provided choices, choice 3: a=1,b=−6,c=4 is the correct one.
Answer
a=1,b=−6,c=4
Exercise #7
y=2x2−5x+6
Video Solution
Step-by-Step Solution
In fact, a quadratic equation is composed as follows:
y = ax²-bx-c
That is,
a is the coefficient of x², in this case 2. b is the coefficient of x, in this case 5. And c is the number without a variable at the end, in this case 6.
Answer
a=2,b=−5,c=6
Exercise #8
y=2x2−3x−6
Video Solution
Step-by-Step Solution
To solve this problem, we'll follow these steps:
Identify the given quadratic function.
Match it with the standard form of a quadratic equation y=ax2+bx+c.
Extract the values of a, b, and c directly from the comparison.
Now, let's work through each step:
Step 1: The given quadratic function is y=2x2−3x−6.
Step 2: The standard form of a quadratic equation is y=ax2+bx+c.
Step 3: By matching the given quadratic function with the standard form:
- The coefficient of x2 is 2, so a=2.
- The coefficient of x is −3, so b=−3.
- The constant term is −6, so c=−6.
Therefore, the solution to the problem is a=2, b=−3, c=−6.
Answer
a=2,b=−3,c=−6
Exercise #9
y=−2x2+3x+10
Video Solution
Step-by-Step Solution
Let's determine the coefficients for the quadratic function given by y=−2x2+3x+10.
Step 1: Identify a.
The coefficient of x2 is −2. Thus, a=−2.
Step 2: Identify b.
The coefficient of x is 3. Thus, b=3.
Step 3: Identify c.
The constant term is 10. Thus, c=10.
Comparing these coefficients to the provided choices, the correct answer is:
a=−2,b=3,c=10.
Therefore, the correct choice is Choice 4.
Answer
a=−2,b=3,c=10
Exercise #10
y=3x2+4x+5
Video Solution
Step-by-Step Solution
To solve this problem, we'll follow these steps:
Step 1: Identify the given quadratic function.
Step 2: Compare it to the standard form y=ax2+bx+c.
Step 3: Determine the values of a, b, and c.
Now, let's work through each step:
Step 1: The problem gives us the quadratic function y=3x2+4x+5.
Step 2: The standard form of a quadratic function is y=ax2+bx+c.
Step 3: By comparing y=3x2+4x+5 with y=ax2+bx+c, we find:
- The coefficient of x2 is a=3.
- The coefficient of x is b=4.
- The constant term is c=5.
Therefore, the solution to the problem is a=3,b=4,c=5.
This matches choice 2, which states: a=3,b=4,c=5.
Answer
a=3,b=4,c=5
Question 1
What is the value of the coefficient \( c \) in the equation below?
\( 3x^2+5x \)
Incorrect
Correct Answer:
0
Question 2
\( y=x^2+x+5 \)
Incorrect
Correct Answer:
\( a=1,b=1,c=5 \)
Question 3
\( y=-x^2+x+5 \)
Incorrect
Correct Answer:
\( a=-1,b=1,c=5 \)
Question 4
\( y=4+3x^2-x \)
Incorrect
Correct Answer:
\( a=3,b=-1,c=4 \)
Question 5
\( y=3x^2+4-5x \)
Incorrect
Correct Answer:
\( a=3,b=-5,c=4 \)
Exercise #11
What is the value of the coefficient c in the equation below?
3x2+5x
Video Solution
Step-by-Step Solution
The quadratic equation of the given problem has already been arranged (that is, all the terms are on one side and 0 is on the other side) thus we can approach the question as follows:
In the problem, the question was asked: what is the value of the coefficientcin the equation?
Let's remember the definition of a coefficient when solving a quadratic equation as well as the formula for the roots:
The rule says that the roots of an equation of the form
ax2+bx+c=0are:
x1,2=2a−b±b2−4ac
That is the coefficient cis the free term - and as such the coefficient of the term is raised to the power of zero -x0(Any number other than zero raised to the power of zero equals 1:
x0=1)
Next we examine the equation of the given problem:
3x2+5x=0Note that there is no free term in the equation, that is, the numerical value of the free term is 0, in fact the equation can be written as follows:
3x2+5x+0=0and therefore the value of the coefficientc is 0.
Hence the correct answer is option c.
Answer
0
Exercise #12
y=x2+x+5
Video Solution
Step-by-Step Solution
To solve this problem, we will identify the parameters of the given quadratic function step-by-step:
Step 1: Define the problem statement: We have y=x2+x+5.
Step 2: Identify the standard form of a quadratic function, which is y=ax2+bx+c.
Step 3: Compare the given quadratic expression with the standard form to identify the coefficients.
Now, let's analyze the quadratic function provided:
From the given expression y=x2+x+5:
- The coefficient of x2 is 1, so a=1.
- The coefficient of x is 1, so b=1.
- The constant term is 5, so c=5.
Therefore, the parameters of the quadratic function are a=1, b=1, and c=5.
Consequently, the correct choice from the provided options is (a=1,b=1,c=5).
Answer
a=1,b=1,c=5
Exercise #13
y=−x2+x+5
Video Solution
Step-by-Step Solution
To solve the problem of identifying the coefficients in the quadratic function y=−x2+x+5, we follow these steps:
Step 1: Write down the general form of a quadratic equation: y=ax2+bx+c.
Step 2: Compare the given equation y=−x2+x+5 to the general form.
Step 3: Identify the value of each coefficient:
The coefficient of x2 is −1, so a=−1.
The coefficient of x is +1, so b=1.
The constant term is +5, so c=5.
Therefore, the parameters of the quadratic function are a=−1, b=1, and c=5.
This matches choice 2, confirming the parameters in the quadratic function.
Final Answer:a=−1,b=1,c=5.
Answer
a=−1,b=1,c=5
Exercise #14
y=4+3x2−x
Video Solution
Step-by-Step Solution
To solve this problem, we will match the given quadratic equation with its standard form:
Step 1: Identify the standard quadratic form as y=ax2+bx+c.
Step 2: Compare with given equation y=4+3x2−x.
Step 3: Determine the coefficients by arranging the equation in standard form.
Let's now perform the steps:
Step 1: The standard quadratic form is y=ax2+bx+c.
Step 2: The given equation is y=4+3x2−x.
Step 3: Rearrange the given equation to match the standard form:
y=3x2−x+4.
Now, directly compare:
a=3
b=−1
c=4
Therefore, the coefficients are correctly identified as a=3,b=−1, and c=4.
The correct answer is: a=3,b=−1,c=4.
Answer
a=3,b=−1,c=4
Exercise #15
y=3x2+4−5x
Video Solution
Step-by-Step Solution
To solve this problem, we'll follow these steps:
Step 1: Rearrange the given quadratic function into the standard form.
Step 2: Identify the coefficients by comparing them with the standard quadratic function.
Now let's work through each step:
Step 1: The given quadratic is y=3x2+4−5x. Rearrange this function to align terms with their degrees: y=3x2−5x+4.
Step 2: Compare this with the standard quadratic form y=ax2+bx+c, where: a=3 (the coefficient of x2), b=−5 (the coefficient of x), c=4 (the constant term).