Intermediate Single-Variable Algebra - Algebra 2

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Question

Evaluate the discriminant, if any:

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Answer

The formula to determine the discriminant is:

To determine the discriminant, we will need to put the equation in standard form:

Add on both sides.

Subtract on both sides.

Reorder the terms on the left.

Divide by two on both sides.

$\frac{x^2-2x-6 }{2}= \frac{2y}{2}$

The equation in standard form is: $y=\frac{1}{2}x^2-x-3$

The coefficients can be determined to calculate the discriminant.

$a=\frac{1}{2}, b=-1, c=-3$

Substitute the values into the formula.

$D=2b^2-4ac = (-1)^2-4(\frac{1}{2})(-3) = 1+6 = 7$

The answer is:

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Question

Determine the discriminant:

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Answer

The formula for the discriminant is:

Given the polynomial in standard format, we can identify the coefficients of the polynomial to substitute into the equation.

Substitute all the numbers into the equation.

The answer is:

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Question

Evaluate the discriminant:

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Answer

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

Determine the coefficients.

Substitute the terms into the formula.

The answer is:

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Question

Factor the polynomial $y = x^{2} + 5x + 6$ .

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Answer

The product of the last two numbers should be 6, while the sum of the products of the inner and outer numbers should be 5x. Factors of six include 1 and 6, and 2 and 3. In this case, our sum is five so the correct choices are 2 and 3. Then, our factored expression is (x + 2)(x + 3). You can check your answer by using FOIL.

y = x2 + 5x + 6

2 * 3 = 6 and 2 + 3 = 5

(x + 2)(x + 3) = x2 + 5x + 6

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Question

Use FOIL to distribute the following:

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Answer

Make sure you keep track of negative signs when doing FOIL, especially when doing the Outer and Inner steps.

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Question

Solve the following equation by completing the square. Use a calculator to determine the answer to the closest hundredth.

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Answer

To solve by completing the square, you should first take the numerical coefficient to the “right side” of the equation:

Then, divide the middle coefficient by 2:

$\frac{8}{2} = 4$

Square that and add it to both sides:

Now, you can easily factor the quadratic:

Take the square root of both sides:

$x+ 4 = \pm\sqrt{28}$

Finish out the solution:

$x=\pm\sqrt{28}-4$

$\sqrt{28} - 4 = 1.29150262212918$

$-\sqrt{28} - 4 = -9.29150262212918$

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Question

Solve the following equation by completing the square. Use a calculator to determine the answer to the closest hundredth.

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Answer

To solve by completing the square, you should first take the numerical coefficient to the “right side” of the equation:

Then, divide the middle coefficient by 2:

$\frac{-12}{2}=-6$

Square that and add it to both sides:

Now, you can easily factor the quadratic:

Take the square root of both sides:

$x- 6 = \pm\sqrt{52}$

Finish out the solution:

$x=\pm\sqrt{52}+6$

$\sqrt{52} + 6 = 13.21110255092798$

$-\sqrt{52} + 6 = -1.21110255092798$

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Question

Solve the following equation by completing the square. Use a calculator to determine the answer to the closest hundredth.

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Answer

To solve by completing the square, you should first take the numerical coefficient to the “right side” of the equation:

Then, divide the middle coefficient by 2:

$\frac{-20}{2}=-10$

Square that and add it to both sides:

Now, you can easily factor the quadratic:

Take the square root of both sides:

$x - 10 = \pm\sqrt{69}$

Finish out the solution:

$x=\pm\sqrt{69}+10$

$\sqrt{69} +10 = 18.30662386291807$

$-\sqrt{69} +10 = 1.69337613708192$

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Question

Find the LCM of the following polynomials:

$2x\left ( x+1 \right )$ , $4x^{2}\left ( x^{2} -1\right )$ , $6\left ( x-1 \right )$

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Answer

LCM of

LCM of $x, x^{2}=x^{2}$

and since $\left ( x^{2} -1 \right )= \left ( x+1 \right )\left ( x-1 \right )$

The LCM $=12x^{2}\left ( x+1 \right )(x-1)$

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Question

Solve the following equation by completing the square. Use a calculator to determine the answer to the closest hundredth.

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Answer

To solve by completing the square, you should first take the numerical coefficient to the “right side” of the equation:

Then, divide the middle coefficient by 2:

$\frac{5}{2}= 2.5$

Square that and add it to both sides:

Now, you can easily factor the quadratic:

Take the square root of both sides:

$x + 2.5 = \pm \sqrt{18.25}$

Finish out the solution:

$x=\pm\sqrt{18.25}-2.5$

$\sqrt{18.25} - 2.5=1.77200187265877$

$-\sqrt{18.25} - 2.5 = -6.77200187265876$

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Question

Use the quadratic formula to solve for . Use a calculator to estimate the value to the closest hundredth.

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Answer

Recall that the quadratic formula is defined as:

$\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

For this question, the variables are as follows:

Substituting these values into the equation, you get:

$\frac{-14 \pm \sqrt{(14)^2 - 4(3)(9)}}{2(3)}$

$\frac{-14 \pm \sqrt{196 - 108}}{6}$

$\frac{-14 \pm \sqrt{88}}{6}$

Use a calculator to determine the final values.

$\frac{14+\sqrt{88}}{6}=3.90$

$\frac{14-\sqrt{88}}{6}=0.77$

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Question

Solve for . Use the quadratic formula to find your solution. Use a calculator to estimate the value to the closest hundredth.

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Answer

Recall that the quadratic formula is defined as:

$\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

For this question, the variables are as follows:

Substituting these values into the equation, you get:

$\frac{-5 \pm \sqrt{(5)^2 - 4(1)(-5)}}{2(1)}$

$\frac{-5 \pm \sqrt{25 +20)}}{2}$

$\frac{-5 \pm \sqrt{45}}{2}$

Use a calculator to determine the final values.

$\frac{-5+\sqrt{45}}{2}=0.85$

$\frac{-5-\sqrt{45}}{2}=-5.85$

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Question

Use FOIL to distribute the following:

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Answer

When the 2 terms differ only in their sign, the -term drops out from the final product.

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Question

Solve the following equation by completing the square. Use a calculator to determine the answer to the closest hundredth.

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Answer

To solve by completing the square, you should first take the numerical coefficient to the “right side” of the equation:

Then, divide the middle coefficient by 2:

$\frac{7}{2} = 3.5$

Square that and add it to both sides:

Now, you can easily factor the quadratic:

Your next step would be to take the square root of both sides. At this point, however, you know that you cannot solve the problem. When you take the square root of both sides, you will be forced to take the square root of . This is impossible (at least in terms of real numbers), meaning that this problem must have no real solution.

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Question

Solve for . Use the quadratic formula to find your solution. Use a calculator to estimate the value to the closest hundredth.

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Answer

Recall that the quadratic formula is defined as:

$\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

For this question, the variables are as follows:

Substituting these values into the equation, you get:

$\frac{-(-17) \pm \sqrt{(-17)^2 - 4(5)(12)}}{2(5)}$

$\frac{17 \pm \sqrt{289 - 240}}{10}$

$\frac{17 \pm \sqrt{49}}{10}$

$\frac{17 \pm 7}{10}$

Separate this expression into two fractions and simplify to determine the final values.

$\frac{17 + 7}{10} = \frac{24}{10} = 2.4$

$\frac{17 - 7}{10} = \frac{10}{10} = 1$

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Question

Evaluate $(2x+3)^{2}$

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Answer

In order to evaluate $(2x+3)^{2}$ one needs to multiply the expression by itself using the laws of FOIL. In the foil method, one multiplies in the following order: first terms, outer terms, inner terms, and last terms.

$\dpi{100} (2x+3)^{2}=(2x+3)(2x+3)$

Multiply terms by way of FOIL method.

Now multiply and simplify.

$=4x^{2}+6x+6x+9$

$\rightarrow 4x^{2}+12x+9$

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Question

Expand this expression:

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Answer

Use the FOIL method (First, Outer, Inner, Last):

Put all of these terms together:

Combine like terms:

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Question

Solve for . Use the quadratic formula to find your solution. Use a calculator to estimate the value to the closest hundredth.

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Answer

Recall that the quadratic formula is defined as:

$\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

For this question, the variables are as follows:

Substituting these values into the equation, you get:

$\frac{-(8) \pm \sqrt{(8)^2 - 4(3)(-11)}}{2(3)}$

$\frac{-8 \pm \sqrt{64 - (-132)}}{6}$

$\frac{-8 \pm \sqrt{64 +132}}{6}$

$\frac{-8 \pm \sqrt{196}}{6}$

$\frac{-8 \pm 14}{6}$

Separate this expression into two fractions and simplify to determine the final values.

$\frac{-8 + 14}{6} = \frac{6}{6} = 1$

$\frac{-8 - 14}{6} = \frac{-22}{6} = -3.67$

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Question

Find the discriminant for the quadratic equation

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Answer

The discriminant is found using the formula . In this case:

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Question

Subtract the following expressions: $\frac{x}{2x+1}-\frac{1}{2x}$

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Answer

In order to subtract the fractions, multiply both denominators together in order to obtain the least common denominator.

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

Simplify the numerators.

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

Combine the numerators.

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

The answer is: $\frac{2x^2-2x-1}{4x^2+2x}$

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