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Algebra II : Algebra II

Study concepts, example questions & explanations for Algebra II

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Example Questions

Example Question #1431 : Algebra Ii

Evaluate the discriminant: y=8x^2-4

Possible Answers:

128

256

Correct answer:

128

Explanation:

Write the formula for the discriminant. The discriminant is the term inside the square root of the quadratic formula.

D=b^2-4ac

Determine the coefficients.

a=8, b=0, c=-4

Substitute the terms into the formula.

D=0^2-4(8)(-4) = 128

The answer is: 128

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Example Question #131 : Understanding Quadratic Equations

What is the discriminant? y=9x^2-10

Possible Answers:

720

360

180

810

Correct answer:

360

Explanation:

The equation is already in the standard form of y=ax^2+bx+c , where the discriminant is defined as:

D=b^2-4ac

Following the coefficients of each term with the polynomial in standard form, we can determine the values of each variable.

a=9,b=0, c=-10

Substitute the terms into the equation.

D=0^2-4(9)(-10) = 360

The answer is: 360

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Example Question #132 : Understanding Quadratic Equations

Identify the discriminant of the parabola given the roots: x=1, x=-3

Possible Answers:

Correct answer:

Explanation:

We will need to use the roots to write the quadratic equation.

y=(x-1)(x+3)

Simplify by using the FOIL method.

y=x^2+3x-1(x)-3 = x^2+2x-3

y= x^2+2x-3

The discriminant of the parabola in standard form is defined as:

D=b^2-4ac

where a=1, b=2, c=-3 .

Substitute the values in the equation.

D=2^2-4(1)(-3) = 4+12 =16

The answer is:

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Example Question #1434 : Algebra Ii

Evaluate the discriminant: y=-3(x^2-x-7)

Possible Answers:

132

261

Correct answer:

261

Explanation:

The given equation is not fully simplified in standard form: y=ax^2+bx+c

Distribute the negative three across each term in the parentheses.

y=-3(x^2-x-7) = -3x^2+3x+21

a=-3, b=3, c=21

Write the formula for the discriminant.

D=b^2-4ac

Substitute the known terms.

D=b^2-4ac = 3^2-4(-3)(21) = 9+252 = 261

The answer is: 261

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Example Question #133 : Understanding Quadratic Equations

Identify the discriminant: y=4x^2-3x-3

Possible Answers:

$\textup{There is no discriminant.}$

Correct answer:

Explanation:

The equation is already in standard form y=ax^2+bx+c .

Following the coefficients:

a=4,b=-3, c=-3

The discriminant is the term inside the square root of the quadratic equation.

D=b^2-4ac

Substitute the values.

D=(-3)^2-4(4)(-3) =57

The discriminant is:

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Example Question #1436 : Algebra Ii

Determine the discriminant if the parabola has $x=\frac{1}{4},\frac{1}{2}$ as roots.

Possible Answers:

$\frac{23}{24}$

$\frac{17}{16}$

$\textup{The answer is not given.}$

$\frac{1}{16}$

$\frac{7}{8}$

Correct answer:

$\frac{1}{16}$

Explanation:

Given the roots, we can set up the binomials of the parabola.

$y=(x-\frac{1}{4})(x-\frac{1}{2})$

Use the FOIL method to simplify these binomials.

$(x)(x)+(x)(-\frac{1}{2})+(-\frac{1}{4})(x)+(-\frac{1}{4})(-\frac{1}{2})$

Simplify all the terms by distribution.

$x^2-\frac{1}{2}x-\frac{1}{4}x-\frac{1}{8}=x^2-\frac{2}{4}x-\frac{1}{4}x+\frac{1}{8}$

The equation of the parabola in standard form is:

$y=x^2-\frac{3}{4}x+\frac{1}{8}$

$a=1, b= -\frac{3}{4}, c=\frac{1}{8}$

The equation of the discriminant is the square root of the term under the radical of the quadratic equation. Substitute the terms into the equation.

$D=b^2-4ac = (-\frac{3}{4})^2-4(1)(\frac{1}{8})$

$\frac{9}{16}-\frac{4}{8} = \frac{9}{16}-\frac{8}{16}$

The answer is: $\frac{1}{16}$

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Example Question #134 : Understanding Quadratic Equations

Determine the discriminant of the parabola: y=5x^2-7x+7

Possible Answers:

189

Correct answer:

Explanation:

Write the formula for the discriminant. This is the term inside the radical of the quadratic equation.

D=b^2-4ac

a=5, b=-7, c=7

Substitute the values into the equation.

D=(-7)^2-4(5)(7) = 49-140 = -91

The answer is:

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Example Question #135 : Understanding Quadratic Equations

Determine the discriminant of the following function: y=-8x^2-7

Possible Answers:

$\textup{There is no discriminant.}$

224

214

Correct answer:

Explanation:

The equation is already in standard form: y=ax^2+bx+c

Identify the coefficients.

a=-8, b=0, c=-7

The formula for the discriminant is:

D=b^2-4ac

Substitute the values into the equation.

D=0^2-4(-8)(-7) = -224

The answer is:

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Example Question #41 : Discriminants

Determine the discriminant: y=-2x^2-6

Possible Answers:

Correct answer:

Explanation:

Write the formula for the discriminant. The discriminant is the term inside the square root portion of the quadratic formula.

D=b^2-4ac

Substitute the known coefficients of the polynomial in standard form:

y=ax^2+bx+c

a=-2, b=0, c=-6

D=0^2-4(-2)(-6) =-48

The answer is:

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Example Question #1440 : Algebra Ii

For the given polynomial function f(x) , find the relationship between and so that the function will have:

a) Two non-zero real roots

b) One non-zero real root

c) Two Complex Roots

$f(x)=x^3+7\sqrt{A}x^2+2Bx$

Where the constants and can be any non-zero, positive real number.

Possible Answers:

a) $A>\frac{49}{8}B$

b) $A=\frac{49}{8}B$

c) $A<\frac{49}{8}B$

a) $\sqrt{A}<\frac{7}{2}B$

b) $\sqrt{A}=\frac{7}{2}B$

c) $\sqrt{A}>\frac{7}{2}B$

a) $A>\frac{8}{49}B$

b) $A=\frac{8}{49}B$

c) $A<\frac{8}{49}B$

a) $\sqrt{A}>\frac{2}{7}B$

b) $\sqrt{A}=\frac{2}{7}B$

c) $\sqrt{A}<\frac{2}{7}B$

a) $A<\frac{49}{8}B$

b) $A=\frac{49}{8}B$

c) $A>\frac{49}{8}B$

Correct answer:

a) $A>\frac{8}{49}B$

b) $A=\frac{8}{49}B$

c) $A<\frac{8}{49}B$

Explanation:

` $f(x)=x^3+7\sqrt{A}x^2+2Bx$

If we were to find the roots of this equation we would first set f(x)=0 and attempt to factor:

$x^3+7\sqrt{A}x^2+2Bx=0$

$x(x^2+7\sqrt{A}x+2B) = 0$

The trivial solution is x=0 , the other possible non-zero solutions will depend on the quadratic factor. A quadratic equation may have either 2 complex roots, 2 real roots, or 1 real root. The conditions determined by a the radical term in the quadratic equation...the discriminate:

For a quadratic equation, or quadratic factor, of the form ax^2+bx+c = 0 the discriminate is defined as:

D = b^2-4ca

For the quadratic factor in this problem we have, $a = 1 \: \: \: \:b=7\sqrt{A}\: \: \: \:c=2B$

The discriminate in this case is then,

D=49A-4(2B)

D=49A-8B

Now let's recall how the discriminate determines the number and types of roots

$D>0 \: \: \: \: \rightarrow\: \: \: \:\; two \: \:real \: \:roots$

$D=0 \: \: \: \: \rightarrow\: \: \: \:\: \: \: one \: \: \: \: real \: \:root$

$D<0 \: \: \: \: \rightarrow\: \: \: \:\:two \: \:complex \: \:roots$

a) Two Real Roots

When the discriminate is greater than zero the function has 2 real-valued roots.

$D>0\: \: \: \: \rightarrow \: \: \: \:49A-8B>0 \: \: \: \: \rightarrow\: \: \: \:A>\frac{8}{49}B$

b) One Real Root

When the discriminate is zero, the function has 1 real valued root:

D = 0

Apply to the discriminate and solve for .

$49A=8B\: \: \: \: \rightarrow \: \: \: \:A=\frac{8}{49}B$

c) Two Complex Roots

Similarly, for two complex roots solve D<0 , we have:

$A<\frac{8}{49}B$

Summary

a) $A>\frac{8}{49}B$

b) $A=\frac{8}{49}B$

c) $A<\frac{8}{49}B$

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Algebra II : Algebra II

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #1431 : Algebra Ii

Example Question #131 : Understanding Quadratic Equations

Example Question #132 : Understanding Quadratic Equations

Example Question #1434 : Algebra Ii

Example Question #133 : Understanding Quadratic Equations

Example Question #1436 : Algebra Ii

Example Question #134 : Understanding Quadratic Equations

Example Question #135 : Understanding Quadratic Equations

Example Question #41 : Discriminants

Example Question #1440 : Algebra Ii

All Algebra II Resources

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