Understanding Quadratic Equations

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Questions 1 - 10

Use the discriminant to determine the nature of the roots:

imaginary roots

imaginary root

real roots

real root

Cannot be determined

Explanation

The formula for the discriminant is:

Since the discriminant is negative, there are imaginary roots.

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

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

$\dpi{100} 4x^{2}+4y^{2}$

$\dpi{100} 4x^{2}+4xy+4y^{2}$

$\dpi{100} x^{2}+xy+y^{2}$

$\dpi{100} 2x^{2}+4xy+2y^{2}$

Explanation

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.

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

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

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

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

$\dpi{100} 4x^{2}+4y^{2}$

$\dpi{100} 4x^{2}+4xy+4y^{2}$

$\dpi{100} x^{2}+xy+y^{2}$

$\dpi{100} 2x^{2}+4xy+2y^{2}$

Explanation

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

Multiply terms by way of FOIL method.

Now multiply and simplify.

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

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

Use the discriminant to determine the nature of the roots:

imaginary roots

imaginary root

real roots

real root

Cannot be determined

Explanation

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:

irrational roots

rational roots

imaginary roots

rational root

imaginary root

Explanation

The formula for the discriminant is:

$=1200^{}$

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

Write an equation with the given roots:

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

Explanation

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 :

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

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

$x^{2}+y^{2}$

$\dpi{100} x^{2}-y^{2}$

$\dpi{100} -x^{2}+2xy-y^{2}$

$\dpi{100} 2x^{2}-4xy+2y^{2}$

Explanation

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.

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

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

Use the discriminant to determine the nature of the roots:

irrational roots

rational roots

imaginary roots

rational root

imaginary root

Explanation

The formula for the discriminant is:

$=1200^{}$

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

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

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

$x^{2}+y^{2}$

$\dpi{100} x^{2}-y^{2}$

$\dpi{100} -x^{2}+2xy-y^{2}$

$\dpi{100} 2x^{2}-4xy+2y^{2}$

Explanation

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

Multiply terms by way of FOIL method.

Now multiply and simplify, paying attention to signs.

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

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

Write an equation with the given roots:

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

Explanation

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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