Algebra II - Math

Card 0 of 1752

Question

Given $2x^{2}+3x-5=0$ , what is the value of the discriminant?

Answer

In general, the discriminant is $b^{2}-4ac$ .

In this particual case .

Plug in these three values and simplify: $(3)^{2}-4(2)(-5)= 49$

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Question

Use the discriminant to determine the nature of the roots:

Answer

The formula for the discriminant is:

Since the discriminant is negative, there are imaginary roots.

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Question

Use the discriminant to determine the nature of the roots:

Answer

The formula for the discriminant is:

$=1200^{}$

Since the discriminant is positive and not a perfect square, there are irrational roots.

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Question

Use the discriminant to determine the nature of the roots:

Answer

The formula for the discriminant is:

Since the discriminant is negative, there are imaginary roots.

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Question

List the transformations that have been enacted upon the following equation:

$f(x)=4[6(x-3)]^{4}-7$

Answer

Since the equation given in the question is based off of the parent function $y = x^{4}$ , we can write the general form for transformations like this:

$g(x) = a[b(x-c)^{4}]+d$

determines the vertical stretch or compression factor.

If $\left | a \right |$ is greater than 1, the function has been vertically stretched (expanded) by a factor of .

If $\left | a \right |$ is between 0 and 1, the function has been vertically compressed by a factor of $\frac{1}{\left | a \right |}$ .

In this case, is 4, so the function has been vertically stretched by a factor of 4.

determines the horizontal stretch or compression factor.

If $\left | b \right |$ is greater than 1, the function has been horizontally compressed by a factor of .

If $\left | b \right |$ is between 0 and 1, the function has been horizontally stretched (expanded) by a factor of $\frac{1}{\left | b \right | }$ .

In this case, is 6, so the function has been horizontally compressed by a factor of 6. (Remember that horizontal stretch and compression are opposite of vertical stretch and compression!)

determines the horizontal translation.

If is positive, the function was translated units right.

If is negative, the function was translated units left.

In this case, is 3, so the function was translated 3 units right.

determines the vertical translation.

If is positive, the function was translated units up.

If is negative, the function was translated units down.

In this case, is -7, so the function was translated 7 units down.

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Question

Find the value of $\sqrt{80} + \sqrt{20}$ .

Answer

To solve this equation, we have to factor our radicals. We do this by finding numbers that multiply to give us the number within the radical.

$\sqrt{80} = \sqrt{20}\sqrt{4}$

$\sqrt{20} = \sqrt{5}\sqrt{4}$

Add them together:

$\sqrt5\sqrt4\sqrt4 + \sqrt5\sqrt4$

4 is a perfect square, so we can find the root:

$(2)(2)\sqrt5 + 2\sqrt5$

$4\sqrt5 + 2\sqrt5 = 6\sqrt5$

Since both have the same radical, we can combine them:

$4\sqrt5 + 2\sqrt5 = 6\sqrt5$

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Question

Simplify the expression:

$\frac{3\sqrt[4]{32}}{2\sqrt[4]{162}}$

Answer

Use the multiplication property of radicals to split the fourth roots as follows:

$\rightarrow \frac{3\sqrt[4]{16}\sqrt[4]{2}}{2\sqrt[4]{81}\sqrt[4]{2}}$

Simplify the new roots:

$\rightarrow \frac{3(2)\sqrt[4]{2}}{2(3)\sqrt[4]{2}}$

$\rightarrow \frac{6\sqrt[4]{2}}{6\sqrt[4]{2}}$

$\rightarrow 1$

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Question

Factor and simplify the following radical expression:

$\sqrt[4]{81x^6y^5z^{10}}$

Answer

Begin by factoring the integer:

$\sqrt[4]{81x^6y^5z^{10}}$

$\sqrt[4]{3^4x^6y^5z^{10}}$

$3\sqrt[4]{x^6y^5z^{10}}$

Now, simplify the exponents:

$3\sqrt[4]{x^4x^2y^4yz^{4}z^4z^2}$

$3xyzz\sqrt[4]{x^2yz^2}$

$3xyz^2\sqrt[4]{x^2yz^2}$

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Question

Factor and simplify the following radical expression:

$\sqrt[3]{(3n-2)^3}$

Answer

Begin by converting the radical into exponent form:

$\sqrt[3]{(3n-2)^3}$

$(3n-2)^3^{(\frac{1}{3})}$

Now, multiply the exponents:

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Question

Factor and simplify the following radical expression:

$\sqrt[4]{18a^3bc^5}\cdot \sqrt[4]{27a^2b^6c}$

Answer

Begin by converting the radical into exponent form:

$\sqrt[4]{18a^3bc^5}\cdot \sqrt[4]{27a^2b^6c}$

$(18a^3bc^5)^{\frac{1}{4}}\cdot ({27a^2b^6c})^{\frac{1}{4}}$

Now, combine the bases:

$(486a^5b^7c^6)^{\frac{1}{4}}$

Simplify the integer:

$(2\cdot 3\cdot 3^4a^5b^7c^6)^{\frac{1}{4}}$

Now, simplify the exponents:

$(2\cdot 3\cdot 3^4aa^4b^3b^4c^2c^4)^{\frac{1}{4}}$

Convert back into radical form and simplify:

$\sqrt[4]{2\cdot 3\cdot 3^4aa^4b^3b^4c^2c^4}$

$3abc\sqrt[4]{2\cdot 3ab^3c^2}$

$3abc\sqrt[4]{6ab^3c^2}$

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Question

Factor and simplify the following radical expression:

$(4-2\sqrt{2})(\sqrt{2}-4)$

Answer

Begin by using the FOIL method (First Outer Inner Last) to expand the expression.

$(4-2\sqrt{2})(\sqrt{2}-4)$

$4\sqrt{2}-16-4+8\sqrt{2}$

Now, combine like terms:

$12\sqrt{2}-20$

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Question

Factor and simplify the following radical expression:

$\frac{3-2\sqrt{3}}{2\sqrt{3}+3}$

Answer

Begin by multiplying the numerator and denominator by the complement of the denominator:

$\frac{3-2\sqrt{3}}{2\sqrt{3}+3} \cdot \frac{2\sqrt{3}-3}{2\sqrt{3}-3}$

Use the FOIL method to multiply the radicals. F (first) O (outer) I (inner) L (last)

$\frac{6\sqrt{3}-9-12+6\sqrt{3}}{12-6\sqrt{3}+6\sqrt{3}-9}$

Now, combine like terms:

$\frac{12\sqrt{3}-21}{3}$

Simplify:

$4\sqrt{3}-7$

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Question

Factor and simplify the following radical expression:

$\sqrt{\frac{1}{5}}-2\sqrt{20}+3\sqrt{5}$

Answer

Begin by factoring the radicals:

$\sqrt{\frac{1}{5}}-2\sqrt{20}+3\sqrt{5}$

$\sqrt{\frac{1}{5}}-4\sqrt{5}+3\sqrt{5}$

Combine like terms:

$\sqrt{\frac{1}{5}}-\sqrt{5}$

Multiply the left side by $\frac{\sqrt{5}}{\sqrt{5}}$ and the right side by $\frac{5}{5}$

$\frac{1}{\sqrt{5}}-\sqrt{5}$

$\frac{1}{\sqrt{5}}\cdot \frac{\sqrt{5}}{\sqrt{5}}-\sqrt{5}\cdot\frac{5}{5}$

$\frac{\sqrt{5}}{5}-\frac{5\sqrt{5}}{5}$

$\frac{-4\sqrt{5}}{5}$

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Question

Factor and simplify the following radical expression:

$\sqrt[4]{24a^2b^3c^5}\cdot\sqrt[4]{48a^3b^3c^6}$

Answer

Begin by converting the radicals into exponent form:

$\sqrt[4]{24a^2b^3c^5}\cdot\sqrt[4]{48a^3b^3c^6}$

$(24a^2b^3c^5)^{\frac{1}{4}}\cdot(48a^3b^3c^6)^{\frac{1}{4}}$

$(3\cdot 2^3a^2b^3c^5)^{\frac{1}{4}}\cdot(3\cdot 2^4a^3b^3c^6)^{\frac{1}{4}}$

Now, combine the bases:

$(3^2\cdot 2^7a^5b^6c^{11})^{\frac{1}{4}}$

$(3^2\cdot 2^7aa^4b^2b^4c^3c^{8})^{\frac{1}{4}}$

Convert back into radical form and simplify:

$\sqrt[4]{3^2\cdot 2^7aa^4b^2b^4c^3c^{8}}$

$2abc^2\sqrt[4]{3^22^3ab^2c^3}$

$2abc^2\sqrt[4]{72ab^2c^3}$

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Question

Factor and simplify the following radical expression:

$\sqrt{3mn}+\sqrt{27m^3n}$

Answer

Begin by simplifying the right side of the rational expression:

$\sqrt{3mn}+\sqrt{27m^3n}$

$\sqrt{3mn}+3m\sqrt{3mn}$

$1\sqrt{3mn}+3m\sqrt{3mn}$

Now, combine like terms:

$(1+3m)\sqrt{3mn}$

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Question

Factor and simplify the following radical expression:

$(4-\sqrt{3})(6-\sqrt{3})$

Answer

Begin by using the FOIL method to multiply the radical expression. F (first) O (outer) I (inner) L (last)

$(4-\sqrt{3})(6-\sqrt{3})$

$24-4\sqrt{3}-6\sqrt{3}+3$

Now, combine like terms:

$27-10\sqrt{3}$

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Question

Factor and simplify the following radical expression:

$\sqrt[3]{\frac{2}{5n}}$

Answer

Begin by multiplying the numerator and denominator by $\frac{\sqrt[3]{(5n)^2}}{\sqrt[3]{(5n)^2}}$ :

$\sqrt[3]{\frac{2}{5n}}$

$\frac{\sqrt[3]{2}}{\sqrt[3]{5n}}\cdot$ $\frac{\sqrt[3]{(5n)^2}}{\sqrt[3]{(5n)^2}}$

$\frac{\sqrt[3]{50n^2}}{5n}$

The expression cannot be further simplified.

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Question

Factor and simplify the following radical expression:

$\frac{2-\sqrt{2}}{4+2\sqrt{2}}$

Answer

Begin by multiplying the numerator and denominator by the complement of the denominator:

$\frac{2-\sqrt{2}}{4+2\sqrt{2}}$

$\frac{2-\sqrt{2}}{4+2\sqrt{2}} \cdot \frac{4-2\sqrt{2}}{4-2\sqrt{2}}$

$\frac{8-4\sqrt{2}-4\sqrt{2}+4}{16-8}$

Combine like terms and simplify:

$\frac{12-8\sqrt{2}}{8}$

$\frac{3-2\sqrt{2}}{2}$

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Question

Factor and simplify the following radical expression:

$\sqrt{\frac{1}{3}}+\sqrt{75}-2\sqrt{3}$

Answer

Begin by factoring the radicals and combining like terms:

$\sqrt{\frac{1}{3}}+\sqrt{75}-2\sqrt{3}$

$\sqrt{\frac{1}{3}}+5\sqrt{3}-2\sqrt{3}$

$\sqrt{\frac{1}{3}}+3\sqrt{3}$

Multiply the left side of the equation by $\frac{\sqrt{3}}{\sqrt{3}}$ and the right side by $\frac{3}{3}$ :

${\frac{1}{\sqrt{3}}}+3\sqrt{3}$

${\frac{1}{\sqrt{3}}}\cdot \frac{\sqrt{3}}{\sqrt{3}}+3\sqrt{3}\cdot\frac{3}{3}$

$\frac{\sqrt{3}}{3}+\frac{9\sqrt{3}}{3}$

$\frac{10\sqrt{3}}{3}$

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Question

Simplify the following radical expression:

$\sqrt[4]{64x^5y^6z^9}$

Answer

Begin by factoring the integer:

$\sqrt[4]{64x^5y^6z^9}$

$\sqrt[4]{2^6x^5y^6z^9}$

Factor the exponents:

$4xyz^2\sqrt[4]{2xy^2z}$

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