Intermediate Single-Variable Algebra - Algebra 2

Card 1 of 2896

0

Didn't Know

0

Didn't Know

Knew It

0

1 of 2019 left

Question

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

Tap to reveal answer

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

← Didn't Know|Knew It →

Question

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

Tap to reveal answer

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:

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

$\sqrt{28} - 4 = 1.29150262212918$

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

← Didn't Know|Knew It →

Question

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

Tap to reveal answer

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:

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

$\sqrt{52} + 6 = 13.21110255092798$

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

← Didn't Know|Knew It →

Question

Use FOIL to distribute the following:

Tap to reveal answer

Answer

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

← Didn't Know|Knew It →

Question

Find the LCM of the following polynomials:

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

Tap to reveal answer

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

← Didn't Know|Knew It →

Question

Use FOIL to distribute the following:

Tap to reveal answer

Answer

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

← Didn't Know|Knew It →

Question

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

Tap to reveal answer

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:

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

$\sqrt{69} +10 = 18.30662386291807$

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

← Didn't Know|Knew It →

Question

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

Tap to reveal answer

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:

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

$\sqrt{18.25} - 2.5=1.77200187265877$

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

← Didn't Know|Knew It →

Question

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

Tap to reveal answer

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.

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

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

← Didn't Know|Knew It →

Question

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

Tap to reveal answer

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.

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

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

← Didn't Know|Knew It →

Question

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

Tap to reveal answer

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.

← Didn't Know|Knew It →

Question

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

Tap to reveal answer

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.

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

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

← Didn't Know|Knew It →

Question

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

Tap to reveal answer

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.

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

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

← Didn't Know|Knew It →

Question

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

Tap to reveal answer

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.

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

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

← Didn't Know|Knew It →

Question

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

Tap to reveal answer

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

← Didn't Know|Knew It →

Question

Expand this expression:

Tap to reveal answer

Answer

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

Put all of these terms together:

Combine like terms:

← Didn't Know|Knew It →

Question

Solve: $\frac{2}{x+3}+\frac{8}{x}$

Tap to reveal answer

Answer

To simplify this expression, we will need to multiply both denominators together to find the least common denominator.

Convert both fractions to the common denominator.

$\frac{2}{x+3}+\frac{8}{x}=\frac{2x}{x^2+3x}+\frac{8(x+3)}{x^2+3x}$

Combine the fractions.

$\frac{2x+8x+24}{x^2+3x} = \frac{10x+24}{x^2+3x}$

The answer is: $\frac{10x+24}{x^2+3x}$

← Didn't Know|Knew It →

Question

Find the discriminant for the quadratic equation

Tap to reveal answer

Answer

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

← Didn't Know|Knew It →

Question

Simplify:

$\frac{4x-2}{2-4x}$

Tap to reveal answer

Answer

When working with a rational expression, you want to first put your monomials in standard format.

Re-order the bottom expression, so it is now reads .

Then factor a out of the expression, giving you .

The new fraction is $\frac{4x-2}{-1(4x-2)}$ .

Divide out the like term, , leaving $\frac{1}{-1}$ , or .

← Didn't Know|Knew It →

Question

Which of the following expressions is a factor of this polynomial: 3x² + 7x – 6?

Tap to reveal answer

Answer

The polynomial factors into (x + 3) (3x - 2).

3x² + 7x – 6 = (a + b)(c + d)

There must be a 3x term to get a 3x² term.

3x² + 7x – 6 = (3x + b)(x + d)

The other two numbers must multiply to –6 and add to +7 when one is multiplied by 3.

b * d = –6 and 3d + b = 7

b = –2 and d = 3

3x² + 7x – 6 = (3x – 2)(x + 3)

(x + 3) is the correct answer.

← Didn't Know|Knew It →

0

Knew It