Understanding Quadratic Equations - Math

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Solve the following equation using the quadratic form:

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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}$$

Solve the following equation using the quadratic form:

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

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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:

Write an equation with the given roots:

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

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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:

Multiply the equation by :

Write an equation with the given roots:

$4 \pm $\sqrt{3}$$

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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:

Write an equation with the given roots:

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

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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:

Multiply the equation by :

Write an equation with the given roots:

$2 \pm $\sqrt{3}$$

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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:

Write an equation with the given roots:

$5 \pm 3i$

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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:

Given , what is the value of the discriminant?

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In general, the discriminant is .

In this particual case .

Plug in these three values and simplify:

Use the discriminant to determine the nature of the roots:

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The formula for the discriminant is:

Since the discriminant is negative, there are imaginary roots.

Use the discriminant to determine the nature of the roots:

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The formula for the discriminant is:

$=1200^{}$

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

Use the discriminant to determine the nature of the roots:

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The formula for the discriminant is:

Since the discriminant is negative, there are imaginary roots.

Solve the equation for .

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

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$\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

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

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

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

Evaluate

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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.

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

Multiply terms by way of FOIL method.

Now multiply and simplify.

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

Evaluate

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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.

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

Evaluate

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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.

Multiply terms by way of FOIL method.

Now multiply and simplify, paying attention to signs.

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

Evaluate

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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.

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

Evaluate

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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.

Multiply terms by way of FOIL method.

Now multiply and simplify, paying attention to signs.

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

FOIL .

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

Simplify to get .

FOIL .

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is the same thing as .

Remember that FOIL stands for First Outer Inner Last.

For this problem, that would be .

Simplify that to .

FOIL .

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Remember FOIL stands for First Outer Inner Last.

Combine like terms to get .