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Algebra II : Understanding Quadratic Equations

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Example Questions

Example Question #1 : Understanding Quadratic Equations

Solve the equation for .

$\small \frac{1}{x}=\frac{x+1}{6}$

Possible Answers:

$\small x=-3\ or\ 2$

$\small x=3\ or\ -2$

$\small x=3\ or\ 2$

$\small x=-3\ or\ -2$

Correct answer:

$\small x=-3\ or\ 2$

Explanation:

$\small \small \frac{1}{x}=\frac{x+1}{6}$

Cross multiply.

$\small 6=x(x+1)$

$\small 6=x^2+x$

Set the equation equal to zero.

$\small 0=x^2+x-6$

Factor to find the roots of the polynomial.

3*-2=-6 and 3+(-2)=1

$\small 0=(x+3)(x-2)$

$\small 0=x+3; x=-3$

$\small 0=x-2; x=2$

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Example Question #1 : How To Use The Grid Method For Foil

$Expand \; (5x^2 +x + 1)(3x-2)$

Possible Answers:

15 x^2-6 x-2

8x^3-7 x^2+x-2

15 x^3+7 x^2-x-2

15 x^3-7 x^2+x-2

Correct answer:

15 x^3-7 x^2+x-2

Explanation:

$Multiply \; (3 x-2) (5 x^2+x+1)\; using \; a \; grid$

$Answer: \; 15 x^3-7 x^2+x-2$

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Example Question #2 : Simplifying And Expanding Quadratics

Solve the equation for :

$\frac{x}{2}=\frac{-1}{x+3}$

Possible Answers:

x=1

x=0

x=1, x=2

x=1, x=0

x=-1, x=-2

Correct answer:

x=-1, x=-2

Explanation:

$\frac{x}{2}=\frac{-1}{x+3}$

1. Cross multiply:

$x^{2}+3x=-2$

2. Set the equation equal to :

$x^{2}+3x+2=0$

3. Factor to find the roots:

(x+2)(x+1)=0

x+2=0 , so x=-2

x+1=0 , so x=-1

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Example Question #2 : Simplifying And Expanding Quadratics

If you were to solve $x^{2}+8x+9=0$ by completing the square, which of the following equations in the form (x+f)^2=g do you get as a result?

Possible Answers:

(x+4)^2=25

(x+4)^2=16

(x+8)^2=16

(x+8)^2=64

(x+4)^2=7

Correct answer:

(x+4)^2=7

Explanation:

When given a quadratic in the form x^2+bx+c=0 and told to solve by completing the square, we start by subtracting from both sides. In this problem is equal to , so we start by subtracting from both sides:

x^2+8x=-9

To complete the square we want to add a number to each side which yields a polynomial on the left side of the equals sign that can be simplified into a squared binomial (x+d)^2 . This number is equal to $(\frac{b}{2})^2$ . In this problem is equal to , so:

$(\frac{b}{2})^2=(\frac{8}{2})^2=4^2=16$

We add to both sides of the equation:

x^2+8x+16=-9+16

We then factor the left side of the equation into binomial squared form and combine like terms on the right:

(x+4)^2=7

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Example Question #3 : Simplifying And Expanding Quadratics

If you were to solve $x^{2}+4x-5=0$ by completing the square, which of the following equations in the form (x+f)^2=g do you get as a result?

Possible Answers:

(x+2)^2=7

(x+2)^2=21

(x+2)^2=5

(x+2)^2=9

(x-2)^2=9

Correct answer:

(x+2)^2=9

Explanation:

x^2+4x=5

To complete the square we want to add a number to each side which yields a polynomial on the left side of the equation that can be simplified into a squared binomial (x+d)^2 . This number is equal to $(\frac{b}{2})^2$ . In this problem is equal to , so:

$(\frac{b}{2})^2=(\frac{4}{2})^2=2^2=4$

We add to both sides of the equation:

x^2+4x+4=5+4

We then factor the left side of the equation into binomial squared form and combine like terms on the right:

(x+2)^2=9

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Example Question #2 : Understanding Quadratic Equations

Expand:

(4x-1)(-2x+3)

Possible Answers:

$8x^{2}+14x-3$

None of the other answers are correct.

$-8x^{2}+14x-3$

$-8x^{2}+10x-3$

$8x^{2}+10x-3$

Correct answer:

$-8x^{2}+14x-3$

Explanation:

Use the FOIL method, which stands for First, Inner, Outer, Last:

(4x)(-2x)+(4x)(3)+(-1)(-2x)+(-1)(3)

= -8x^2+12x+2x-3

$=-8x^{2}+14x-3$

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Example Question #33 : Polynomial Operations

Multiply: $(x^{2} + 3) (4x - 1)$

Possible Answers:

$4x^{3} -x^{2} -12x + 3$

$4x^{3} - 3$

$4x^{3} -x^{2} +12x - 3$

$4x^{3} +x^{2} -12x - 3$

$4x^{3} + 3$

Correct answer:

$4x^{3} -x^{2} +12x - 3$

Explanation:

$(x^{2} + 3) (4x - 1)$

$= x^{2} (4x - 1) + 3(4x - 1)$

$= x^{2} \cdot 4x - x^{2} \cdot 1 + 3 \cdot 4x - 3 \cdot 1$

$= 4x^{3} - x^{2} + 12x - 3$

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Example Question #2 : Simplifying And Expanding Quadratics

Multiply: $(x + 4) (x^{2} -4x + 16)$

Possible Answers:

$x^{3}- 12x - 64$

$x^{3}- 20x + 64$

$x^{3}- 12x + 64$

$x^{3} - 64$

$x^{3} + 64$

Correct answer:

$x^{3} + 64$

Explanation:

$(x + 4) (x^{2} -4x + 16)$

$= x (x^{2} -4x + 16)+ 4 (x^{2} -4x + 16)$

$= x \cdot x^{2} -x \cdot 4x + x \cdot 16+ 4 \cdot x^{2} -4 \cdot 4x + 4 \cdot 16$

$= x^{3} -4x^{2} + 16x+ 4 x^{2} -16x + 64$

$= x^{3} -4x^{2} + 4 x^{2} + 16x -16x + 64$

$= x^{3} + 64$

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Example Question #1 : How To Subtract Trinomials

Subtract:

(x^2+15x-3)-(x^2-5x-3)

Possible Answers:

2x^2+20x+6

20x

20x+6

x^4+20x^2+6

20x^2

Correct answer:

20x

Explanation:

When subtracting trinomials, first distribute the negative sign to the expression being subtracted, and then remove the parentheses: (x^2+15x-3)-(x^2-5x-3)=x^2+15x-3-x^2+5x+3

Next, identify and group the like terms in order to combine them: (x^2-x^2)+(15x+5x)+(3-3)=20x .

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Example Question #2 : How To Multiply Trinomials

Evaluate the following:

(x^2+2x-4)(2x+5)

Possible Answers:

2x^3+9x^2+18x-20

4x^3+6x^2+2x-20

2x^3+9x^2+2x-20

2x^3+9x^2+2x+20

Correct answer:

2x^3+9x^2+2x-20

Explanation:

When multiplying this trinomial by this binomial, you'll need to use a modified form of FOIL, by which every term in the binomial gets multiplied by every term in the trinomial. One way to do this is to use the grid method.

You can also solve it piece-by-piece the way it is set up. First, multiply each of the three terms in the trinomail by . Then multiply each of those three terms again, this time by .

$(x^2+2x-4) \times 2x = 2x^3 + 4x^2 - 8x$

$(x^2+2x-4) \times 5 = 5x^2 + 10x - 20$

Finally, you can combine like terms after this multiplication to get your final simplified answer:

2x^3 + 9x^2 +2x-20

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Algebra II : Understanding Quadratic Equations

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #1 : Understanding Quadratic Equations

Example Question #1 : How To Use The Grid Method For Foil

Example Question #2 : Simplifying And Expanding Quadratics

Example Question #2 : Simplifying And Expanding Quadratics

Example Question #3 : Simplifying And Expanding Quadratics

Example Question #2 : Understanding Quadratic Equations

Example Question #33 : Polynomial Operations

Example Question #2 : Simplifying And Expanding Quadratics

Example Question #1 : How To Subtract Trinomials

Example Question #2 : How To Multiply Trinomials

All Algebra II Resources

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