Algebra II : Quadratic Equations and Inequalities

Study concepts, example questions & explanations for Algebra II

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Example Questions

Example Question #1 : Using Foil

Solve the equation for \(\displaystyle x\).

\(\displaystyle \small \frac{1}{x}=\frac{x+1}{6}\)

Possible Answers:

\(\displaystyle \small x=3\ or\ -2\)

\(\displaystyle \small x=-3\ or\ 2\)

\(\displaystyle \small x=3\ or\ 2\)

\(\displaystyle \small x=-3\ or\ -2\)

Correct answer:

\(\displaystyle \small x=-3\ or\ 2\)

Explanation:

\(\displaystyle \small \small \frac{1}{x}=\frac{x+1}{6}\)

Cross multiply.

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

\(\displaystyle \small 6=x^2+x\)

Set the equation equal to zero.

\(\displaystyle \small 0=x^2+x-6\)

Factor to find the roots of the polynomial.

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

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

\(\displaystyle \small 0=x+3; x=-3\)

\(\displaystyle \small 0=x-2; x=2\)

Example Question #1 : Simplifying And Expanding Quadratics

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

Possible Answers:

\(\displaystyle 15 x^2-6 x-2\)

\(\displaystyle 15 x^3+7 x^2-x-2\)

\(\displaystyle 15 x^3-7 x^2+x-2\)

\(\displaystyle 8x^3-7 x^2+x-2\)

Correct answer:

\(\displaystyle 15 x^3-7 x^2+x-2\)

Explanation:

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

           

\(\displaystyle 5x^2\)

\(\displaystyle x\)

\(\displaystyle 1\)

\(\displaystyle 3x\)

\(\displaystyle 15x^3\)      

\(\displaystyle 3x^2\)     

\(\displaystyle 3x\)

\(\displaystyle -2\)

\(\displaystyle -10x^2\)

\(\displaystyle -2x\)

\(\displaystyle -2\)

 

 

\(\displaystyle Answer: \; 15 x^3-7 x^2+x-2\)

Example Question #2 : Simplifying And Expanding Quadratics

Solve the equation for \(\displaystyle x\):

\(\displaystyle \frac{x}{2}=\frac{-1}{x+3}\)

Possible Answers:

\(\displaystyle x=-1, x=-2\)

\(\displaystyle x=0\)

\(\displaystyle x=1, x=0\)

\(\displaystyle x=1\)

\(\displaystyle x=1, x=2\)

Correct answer:

\(\displaystyle x=-1, x=-2\)

Explanation:

\(\displaystyle \frac{x}{2}=\frac{-1}{x+3}\)

 

1. Cross multiply:

\(\displaystyle x^{2}+3x=-2\)

 

2. Set the equation equal to \(\displaystyle 0\):

\(\displaystyle x^{2}+3x+2=0\)

 

3. Factor to find the roots:

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

\(\displaystyle x+2=0\),  so  \(\displaystyle x=-2\)

\(\displaystyle x+1=0\), so  \(\displaystyle x=-1\)

Example Question #1 : Understanding Quadratic Equations

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

Possible Answers:

\(\displaystyle (x+4)^2=7\)

\(\displaystyle (x+8)^2=16\)

\(\displaystyle (x+4)^2=25\)

\(\displaystyle (x+4)^2=16\)

\(\displaystyle (x+8)^2=64\)

Correct answer:

\(\displaystyle (x+4)^2=7\)

Explanation:

When given a quadratic in the form \(\displaystyle x^2+bx+c=0\) and told to solve by completing the square, we start by subtracting \(\displaystyle c\) from both sides. In this problem \(\displaystyle c\) is equal to \(\displaystyle 9\), so we start by subtracting \(\displaystyle 9\) from both sides:

\(\displaystyle x^2+8x=-9\)

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 \(\displaystyle (x+d)^2\). This number is equal to \(\displaystyle (\frac{b}{2})^2\). In this problem \(\displaystyle b\) is equal to \(\displaystyle 8\), so: 

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

We add \(\displaystyle 16\) to both sides of the equation:

\(\displaystyle x^2+8x+16=-9+16\)

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

\(\displaystyle (x+4)^2=7\)

Example Question #3 : Simplifying And Expanding Quadratics

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

Possible Answers:

\(\displaystyle (x+2)^2=7\)

\(\displaystyle (x-2)^2=9\)

\(\displaystyle (x+2)^2=9\)

\(\displaystyle (x+2)^2=21\)

\(\displaystyle (x+2)^2=5\)

Correct answer:

\(\displaystyle (x+2)^2=9\)

Explanation:

When given a quadratic in the form \(\displaystyle x^2+bx+c=0\) and told to solve by completing the square, we start by subtracting \(\displaystyle c\) from both sides. In this problem \(\displaystyle c\) is equal to \(\displaystyle -5\), so we start by subtracting \(\displaystyle -5\) from both sides:

\(\displaystyle x^2+4x=5\)

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 \(\displaystyle (x+d)^2\). This number is equal to \(\displaystyle (\frac{b}{2})^2\). In this problem \(\displaystyle b\) is equal to \(\displaystyle 4\), so: 

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

We add \(\displaystyle 4\) to both sides of the equation:

\(\displaystyle x^2+4x+4=5+4\)

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

\(\displaystyle (x+2)^2=9\)

Example Question #1 : Quadratic Equations And Inequalities

Expand:

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

Possible Answers:

\(\displaystyle 8x^{2}+10x-3\)

\(\displaystyle -8x^{2}+14x-3\)

\(\displaystyle 8x^{2}+14x-3\)

\(\displaystyle -8x^{2}+10x-3\)

None of the other answers are correct.

Correct answer:

\(\displaystyle -8x^{2}+14x-3\)

Explanation:

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

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

\(\displaystyle = -8x^2+12x+2x-3\)

\(\displaystyle =-8x^{2}+14x-3\)

 

Example Question #2 : Simplifying And Expanding Quadratics

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

Possible Answers:

\(\displaystyle 4x^{3} - 3\)

\(\displaystyle 4x^{3} -x^{2} -12x + 3\)

\(\displaystyle 4x^{3} +x^{2} -12x - 3\)

\(\displaystyle 4x^{3} -x^{2} +12x - 3\)

\(\displaystyle 4x^{3} + 3\)

Correct answer:

\(\displaystyle 4x^{3} -x^{2} +12x - 3\)

Explanation:

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

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

\(\displaystyle = x^{2} \cdot 4x - x^{2} \cdot 1 + 3 \cdot 4x - 3 \cdot 1\)

\(\displaystyle = 4x^{3} - x^{2} + 12x - 3\)

Example Question #5 : Simplifying And Expanding Quadratics

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

Possible Answers:

\(\displaystyle x^{3} + 64\)

\(\displaystyle x^{3}- 20x + 64\)

\(\displaystyle x^{3} - 64\)

\(\displaystyle x^{3}- 12x + 64\)

\(\displaystyle x^{3}- 12x - 64\)

Correct answer:

\(\displaystyle x^{3} + 64\)

Explanation:

\(\displaystyle (x + 4) (x^{2} -4x + 16)\)

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

\(\displaystyle = x \cdot x^{2} -x \cdot 4x + x \cdot 16+ 4 \cdot x^{2} -4 \cdot 4x + 4 \cdot 16\)

\(\displaystyle = x^{3} -4x^{2} + 16x+ 4 x^{2} -16x + 64\)

\(\displaystyle = x^{3} -4x^{2} + 4 x^{2} + 16x -16x + 64\)

\(\displaystyle = x^{3} + 64\)

Example Question #2 : Simplifying And Expanding Quadratics

Subtract:

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

Possible Answers:

\(\displaystyle 2x^2+20x+6\)

\(\displaystyle 20x^2\)

\(\displaystyle 20x+6\)

\(\displaystyle x^4+20x^2+6\)

\(\displaystyle 20x\)

Correct answer:

\(\displaystyle 20x\)

Explanation:

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

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

Example Question #2 : Understanding Quadratic Equations

Evaluate the following:

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

Possible Answers:

\(\displaystyle 2x^3+9x^2+2x+20\)

\(\displaystyle 2x^3+9x^2+18x-20\)

\(\displaystyle 2x^3+9x^2+2x-20\)

\(\displaystyle 4x^3+6x^2+2x-20\)

Correct answer:

\(\displaystyle 2x^3+9x^2+2x-20\)

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 \(\displaystyle 2x\). Then multiply each of those three terms again, this time by \(\displaystyle 5\).

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

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

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

\(\displaystyle 2x^3 + 9x^2 +2x-20\)

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