Quadratic Equations and Inequalities - Math

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Question

Solve the quadratic equation using any method:

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Answer

Use the quadratic formula to solve:

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

$x = \frac{-(c) \pm \sqrt{(c)^2-4(a)(b)}}{2(b)}$

$x = \frac{-c \pm \sqrt{c^2-4ab}}{2b}$

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Question

Solve the following equation using the quadratic form:

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Answer

Factor the equation and solve:

$x = \pm \sqrt{3}$

or

$x = \pm i\sqrt{3}$

Therefore there are four answers:

$x=\pm\sqrt{3}, \pm i\sqrt{3}$

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Question

Solve the following equation using the quadratic form:

$x-4\sqrt{x}-45=0$

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Answer

Factor the equation and solve:

$x-4\sqrt{x}-45=0$

$(\sqrt{x}-9)(\sqrt{x}+5)=0$

$\sqrt{x}=9$

or

$\sqrt{x}=-5$

This has no solutions.

Therefore there is only one answer:

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Question

Write an equation with the given roots:

$-\frac{3}{5}, \frac{1}{2}$

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Answer

To write an equation, find the sum and product of the roots. The sum is the negative coefficient of , and the product is the integer.

Sum: $(\frac{-3}{5})+(\frac{1}{2})=\frac{-1}{10}$

Product: $(\frac{-3}{5})\cdot (\frac{1}{2})=\frac{-3}{10}$

Subtract the sum and add the product.

The equation is:

$x^2 + \frac{1}{10}x-\frac{3}{10}=0$

Multiply the equation by :

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Question

Write an equation with the given roots:

$4 \pm \sqrt{3}$

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Answer

To write an equation, find the sum and product of the roots. The sum is the negative coefficient of , and the product is the integer.

Sum: $(4 + \sqrt{3})+(4 - \sqrt{3})=8$

Product: $(4 + \sqrt{3})\cdot (4 - \sqrt{3})=16-3=13$

Subtract the sum and add the product.

The equation is:

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Question

Write an equation with the given roots:

$\frac{3}{4}, \frac{1}{3}$

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Answer

To write an equation, find the sum and product of the roots. The sum is the negative coefficient of , and the product is the integer.

Sum: $(\frac{3}{4})+(\frac{1}{3})=\frac{13}{12}$

Product: $(\frac{3}{4})\cdot (\frac{1}{3})=\frac{1}{4}$

Subtract the sum and add the product.

The equation is:

$x^2 - \frac{13}{12}x+\frac{1}{4}=0$

Multiply the equation by :

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Question

Write an equation with the given roots:

$2 \pm \sqrt{3}$

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Answer

To write an equation, find the sum and product of the roots. The sum is the negative coefficient of , and the product is the integer.

Sum: $(2 + \sqrt{3})+(2 - \sqrt{3})=4$

Product: $(2 + \sqrt{3})\cdot (2 - \sqrt{3})=4-3=1$

Subtract the sum and add the product.

The equation is:

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Question

Write an equation with the given roots:

$5 \pm 3i$

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Answer

To write an equation, find the sum and product of the roots. The sum is the negative coefficient of , and the product is the integer.

Sum:

Product: $(5+3i)\cdot (5-3i)=25-9i^2=34$

Subtract the sum and add the product.

The equation is:

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Question

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

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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:

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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:

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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:

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Answer

The formula for the discriminant is:

Since the discriminant is negative, there are imaginary roots.

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Question

Solve the equation for .

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

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Answer

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

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.

and

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

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

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

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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

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

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Answer

In order to evaluate $\dpi{100} (2x+2y)^{2}$ 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.

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

Multiply terms by way of FOIL method.

Now multiply and simplify.

$=4x^{2}+4xy+4xy+4y^{2}$

$\rightarrow 4x^{2}+8xy+4y^{2}$

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Question

Evaluate $(x-2)^{2}$

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Answer

In order to evaluate $\dpi{100} \dpi{100} (x-2)^{2}$ 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. Be sure to pay attention to signs.

$(x-2)^{2}=(x-2)(x-2)$

Multiply terms by way of FOIL method.

Now multiply and simplify, paying attention to signs.

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

$\rightarrow x^{2}-4x+4$

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Question

Evaluate

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Answer

In order to evaluate 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. Be sure to pay attention to signs.

Multiply terms by way of FOIL method.

Now multiply and simplify, paying attention to signs.

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

$\rightarrow 2x^{2}+3xy-2y^{2}$

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Question

Evaluate $(x-y)^{2}$

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Answer

In order to evaluate $\dpi{100} \dpi{100} (x-y)^{2}$ 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. Be sure to pay attention to signs.

$(x-y)^{2}=(x-y)(x-y)$

Multiply terms by way of FOIL method.

Now multiply and simplify, paying attention to signs.

$=x^{2}+(-xy)+(-xy)+(y^{2})$

$\rightarrow x^{2}-2xy+y^{2}$

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Question

FOIL .

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Answer

Remember FOIL stands for First Outer Inner Last. That means we can take and turn it into .

Simplify to get .

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Question

FOIL .

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Answer

is the same thing as .

Remember that FOIL stands for First Outer Inner Last.

For this problem, that would be .

Simplify that to .

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