Algebra Examples

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

Roots (also called zeros) are the points where the graph intercepts with the x-axis $(y = 0)$ .

$y = 0$ at the roots

The root at $x = \frac{1}{2}$ was found by solving for $x$ when $(x - (\frac{1}{2})) = y$ and $y = 0$ ( $y = 0$ because the root is the x-intercept).

The factor is $x - \frac{1}{2}$

The root at $x = 3$ was found by solving for $x$ when $(x - (3)) = y$ and $y = 0$ ( $y = 0$ because the root is the x-intercept).

The factor is $x - 3$

Combine all the factors into a single equation.

$y = (x - \frac{1}{2}) (x - 3)$

Multiply all the factors to simplify the equation $y = (x - \frac{1}{2}) (x - 3)$ .

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Expand $(x - \frac{1}{2}) (x - 3)$ using the FOIL Method.

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Apply the distributive property.

$y = x (x - 3) - \frac{1}{2} \cdot (x - 3)$

Apply the distributive property.

$y = x x + x \cdot - 3 - \frac{1}{2} \cdot (x - 3)$

Apply the distributive property.

$y = x x + x \cdot - 3 + (- \frac{1}{2} x - \frac{1}{2} \cdot - 3)$

Remove parentheses.

$y = x x + x \cdot - 3 - \frac{1}{2} x - \frac{1}{2} \cdot - 3$

Simplify and combine like terms.

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Simplify each term.

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Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = x^{1 + 1} + x \cdot - 3 - \frac{1}{2} x - \frac{1}{2} \cdot - 3$

Add $1$ and $1$ to get $2$ .

$y = x^{2} + x \cdot - 3 - \frac{1}{2} x - \frac{1}{2} \cdot - 3$

Move $- 3$ to the left of the expression $x \cdot - 3$ .

$y = x^{2} - 3 \cdot x - \frac{1}{2} x - \frac{1}{2} \cdot - 3$

Multiply $- 3$ by $x$ to get $- 3 x$ .

$y = x^{2} - 3 x - \frac{1}{2} x - \frac{1}{2} \cdot - 3$

Simplify $- \frac{1}{2} x$ .

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Write $x$ as a fraction with denominator $1$ .

$y = x^{2} - 3 x - (\frac{x}{1} \cdot \frac{1}{2}) - \frac{1}{2} \cdot - 3$

Multiply $\frac{x}{1}$ and $\frac{1}{2}$ to get $\frac{x}{2}$ .

$y = x^{2} - 3 x - \frac{x}{2} - \frac{1}{2} \cdot - 3$

Simplify $- \frac{1}{2} \cdot - 3$ .

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Multiply $- 3$ by $- 1$ to get $3$ .

$y = x^{2} - 3 x - \frac{x}{2} + 3 (\frac{1}{2})$

Write $3$ as a fraction with denominator $1$ .

$y = x^{2} - 3 x - \frac{x}{2} + \frac{3}{1} \cdot \frac{1}{2}$

Multiply $\frac{3}{1}$ and $\frac{1}{2}$ to get $\frac{3}{2}$ .

$y = x^{2} - 3 x - \frac{x}{2} + \frac{3}{2}$

To write $\frac{- 3 x}{1}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = x^{2} + (\frac{- 3 x}{1} \cdot \frac{2}{2} - \frac{x}{2}) + \frac{3}{2}$

Write each expression with a common denominator of $2$ , by multiplying each by an appropriate factor of $1$ .

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Combine.

$y = x^{2} + (\frac{- 3 x \cdot 2}{1 \cdot 2} - \frac{x}{2}) + \frac{3}{2}$

Multiply $2$ by $1$ to get $2$ .

$y = x^{2} + (\frac{- 3 x \cdot 2}{2} - \frac{x}{2}) + \frac{3}{2}$

Combine the numerators over the common denominator.

$y = x^{2} + \frac{- 3 x \cdot 2 - x}{2} + \frac{3}{2}$

To write $\frac{x^{2}}{1}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{x^{2}}{1} \cdot \frac{2}{2} + \frac{- 3 x \cdot 2 - x}{2} + \frac{3}{2}$

Write each expression with a common denominator of $2$ , by multiplying each by an appropriate factor of $1$ .

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Combine.

$y = \frac{x^{2} \cdot 2}{1 \cdot 2} + \frac{- 3 x \cdot 2 - x}{2} + \frac{3}{2}$

Multiply $2$ by $1$ to get $2$ .

$y = \frac{x^{2} \cdot 2}{2} + \frac{- 3 x \cdot 2 - x}{2} + \frac{3}{2}$

Combine the numerators over the common denominator.

$y = \frac{x^{2} \cdot 2 + (- 3 x \cdot 2 - x)}{2} + \frac{3}{2}$

Combine the numerators over the common denominator.

$y = \frac{x^{2} \cdot 2 + (- 3 x \cdot 2 - x) + 3}{2}$

Simplify the numerator.

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Move $2$ to the left of the expression $x^{2} \cdot 2$ .

$y = \frac{2 \cdot x^{2} + (- 3 x \cdot 2 - x) + 3}{2}$

Multiply $2$ by $x^{2}$ to get $2 x^{2}$ .

$y = \frac{2 x^{2} + (- 3 x \cdot 2 - x) + 3}{2}$

Multiply $2$ by $- 3$ to get $- 6$ .

$y = \frac{2 x^{2} + (- 6 x - x) + 3}{2}$

Remove parentheses.

$y = \frac{2 x^{2} - 6 x - x + 3}{2}$

Subtract $x$ from $- 6 x$ to get $- 7 x$ .

$y = \frac{2 x^{2} - 7 x + 3}{2}$

Factor by grouping.

$y = \frac{(x - 3) (2 x - 1)}{2}$

Expand $(x - 3) (2 x - 1)$ using the FOIL Method.

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Apply the distributive property.

$y = \frac{x (2 x - 1) - 3 (2 x - 1)}{2}$

Apply the distributive property.

$y = \frac{x (2 x) + x \cdot - 1 - 3 (2 x - 1)}{2}$

Apply the distributive property.

$y = \frac{x (2 x) + x \cdot - 1 + (- 3 (2 x) - 3 \cdot - 1)}{2}$

Remove parentheses.

$y = \frac{x (2 x) + x \cdot - 1 - 3 (2 x) - 3 \cdot - 1}{2}$

Simplify and combine like terms.

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Simplify each term.

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Move $x$ .

$y = \frac{2 (x x) + x \cdot - 1 - 3 (2 x) - 3 \cdot - 1}{2}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \frac{2 x^{1 + 1} + x \cdot - 1 - 3 (2 x) - 3 \cdot - 1}{2}$

Add $1$ and $1$ to get $2$ .

$y = \frac{2 x^{2} + x \cdot - 1 - 3 (2 x) - 3 \cdot - 1}{2}$

Move $- 1$ to the left of the expression $x \cdot - 1$ .

$y = \frac{2 x^{2} - 1 \cdot x - 3 (2 x) - 3 \cdot - 1}{2}$

Multiply $- 1$ by $x$ to get $- 1 x$ .

$y = \frac{2 x^{2} - 1 x - 3 (2 x) - 3 \cdot - 1}{2}$

Rewrite $- 1 x$ as $- x$ .

$y = \frac{2 x^{2} - x - 3 (2 x) - 3 \cdot - 1}{2}$

Multiply $2$ by $- 3$ to get $- 6$ .

$y = \frac{2 x^{2} - x - 6 x - 3 \cdot - 1}{2}$

Multiply $- 3$ by $- 1$ to get $3$ .

$y = \frac{2 x^{2} - x - 6 x + 3}{2}$

Subtract $6 x$ from $- x$ to get $- 7 x$ .

$y = \frac{2 x^{2} - 7 x + 3}{2}$

Split the fraction $\frac{2 x^{2} - 7 x + 3}{2}$ into two fractions.

$y = \frac{2 x^{2} - 7 x}{2} + \frac{3}{2}$

Split the fraction $\frac{2 x^{2} - 7 x}{2}$ into two fractions.

$y = \frac{2 x^{2}}{2} + \frac{- 7 x}{2} + \frac{3}{2}$

Reduce the expression by cancelling the common factors.

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Cancel the common factor.

$y = \frac{2 x^{2}}{2} + \frac{- 7 x}{2} + \frac{3}{2}$

Divide $x^{2}$ by $1$ to get $x^{2}$ .

$y = x^{2} + \frac{- 7 x}{2} + \frac{3}{2}$

Simplify the expression.

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Move the negative in front of the fraction.

$y = x^{2} - \frac{(7) x}{2} + \frac{3}{2}$

Multiply $7$ by $x$ to get $7 x$ .

$y = x^{2} - \frac{7 x}{2} + \frac{3}{2}$

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