Find the Quadratic Equation Given the Roots 4 , -1/2

$4$ , $- \frac{1}{2}$

Step 1

$x = 4$ and $x = - \frac{1}{2}$ are the two real distinct solutions for the quadratic equation, which means that $x - 4$ and $x + \frac{1}{2}$ are the factors of the quadratic equation.

$(x - 4) (x + \frac{1}{2}) = 0$

Step 2

Expand $(x - 4) (x + \frac{1}{2})$ using the FOIL Method.

Apply the distributive property.

$x (x + \frac{1}{2}) - 4 (x + \frac{1}{2}) = 0$

Apply the distributive property.

$x \cdot x + x (\frac{1}{2}) - 4 (x + \frac{1}{2}) = 0$

Apply the distributive property.

$x \cdot x + x (\frac{1}{2}) - 4 x - 4 (\frac{1}{2}) = 0$

Step 3

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x^{2} + x (\frac{1}{2}) - 4 x - 4 (\frac{1}{2}) = 0$

Combine $x$ and $\frac{1}{2}$ .

$x^{2} + \frac{x}{2} - 4 x - 4 (\frac{1}{2}) = 0$

Cancel the common factor of $2$ .

Factor $2$ out of $- 4$ .

$x^{2} + \frac{x}{2} - 4 x + 2 (- 2) (\frac{1}{2}) = 0$

Cancel the common factor.

$x^{2} + \frac{x}{2} - 4 x + 2 \cdot (- 2 (\frac{1}{2})) = 0$

Rewrite the expression.

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

To write $- 4 x$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x^{2} + \frac{x}{2} - 4 x \cdot \frac{2}{2} - 2 = 0$

Combine $- 4 x$ and $\frac{2}{2}$ .

$x^{2} + \frac{x}{2} + \frac{- 4 x \cdot 2}{2} - 2 = 0$

Combine the numerators over the common denominator.

$x^{2} + \frac{x - 4 x \cdot 2}{2} - 2 = 0$

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

$x^{2} \cdot \frac{2}{2} + \frac{x - 4 x \cdot 2}{2} - 2 = 0$

Combine $x^{2}$ and $\frac{2}{2}$ .

$\frac{x^{2} \cdot 2}{2} + \frac{x - 4 x \cdot 2}{2} - 2 = 0$

Combine the numerators over the common denominator.

$\frac{x^{2} \cdot 2 + x - 4 x \cdot 2}{2} - 2 = 0$

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

$\frac{x^{2} \cdot 2 + x - 4 x \cdot 2}{2} - 2 \cdot \frac{2}{2} = 0$

Combine $- 2$ and $\frac{2}{2}$ .

$\frac{x^{2} \cdot 2 + x - 4 x \cdot 2}{2} + \frac{- 2 \cdot 2}{2} = 0$

Combine the numerators over the common denominator.

$\frac{x^{2} \cdot 2 + x - 4 x \cdot 2 - 2 \cdot 2}{2} = 0$

Step 4

Simplify the numerator.

Reorder terms.

$\frac{2 x^{2} - 4 \cdot (2 x) + x - 2 \cdot 2}{2} = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(2 x^{2} - 4 \cdot (2 x)) + x - 2 \cdot 2}{2} = 0$

Factor out the greatest common factor (GCF) from each group.

$\frac{2 x (x - 2 \cdot 2) + 1 (x - 2 \cdot 2)}{2} = 0$

Factor the polynomial by factoring out the greatest common factor, $x - 2 \cdot 2$ .

$\frac{(x - 2 \cdot 2) (2 x + 1)}{2} = 0$

Multiply $- 2$ by $2$ .

$\frac{(x - 4) (2 x + 1)}{2} = 0$

Step 5

Expand $(x - 4) (2 x + 1)$ using the FOIL Method.

Apply the distributive property.

$\frac{x (2 x + 1) - 4 (2 x + 1)}{2} = 0$

Apply the distributive property.

$\frac{x (2 x) + x \cdot 1 - 4 (2 x + 1)}{2} = 0$

Apply the distributive property.

$\frac{x (2 x) + x \cdot 1 - 4 (2 x) - 4 \cdot 1}{2} = 0$

Step 6

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$\frac{2 x \cdot x + x \cdot 1 - 4 (2 x) - 4 \cdot 1}{2} = 0$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$\frac{2 (x \cdot x) + x \cdot 1 - 4 (2 x) - 4 \cdot 1}{2} = 0$

Multiply $x$ by $x$ .

$\frac{2 x^{2} + x \cdot 1 - 4 (2 x) - 4 \cdot 1}{2} = 0$

Multiply $x$ by $1$ .

$\frac{2 x^{2} + x - 4 (2 x) - 4 \cdot 1}{2} = 0$

Multiply $2$ by $- 4$ .

$\frac{2 x^{2} + x - 8 x - 4 \cdot 1}{2} = 0$

Multiply $- 4$ by $1$ .

$\frac{2 x^{2} + x - 8 x - 4}{2} = 0$

Subtract $8 x$ from $x$ .

$\frac{2 x^{2} - 7 x - 4}{2} = 0$

Step 7

Split the fraction $\frac{2 x^{2} - 7 x - 4}{2}$ into two fractions.

$\frac{2 x^{2} - 7 x}{2} + \frac{- 4}{2} = 0$

Step 8

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

$\frac{2 x^{2}}{2} + \frac{- 7 x}{2} + \frac{- 4}{2} = 0$

Step 9

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x^{2}}{2} + \frac{- 7 x}{2} + \frac{- 4}{2} = 0$

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

$x^{2} + \frac{- 7 x}{2} + \frac{- 4}{2} = 0$

Step 10

Move the negative in front of the fraction.

$x^{2} - \frac{7 x}{2} + \frac{- 4}{2} = 0$

Step 11

Divide $- 4$ by $2$ .

$x^{2} - \frac{7 x}{2} - 2 = 0$

Step 12

The standard quadratic equation using the given set of solutions ${4, - \frac{1}{2}}$ is $y = x^{2} - \frac{7 x}{2} - 2$ .

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

Step 13

Find the Quadratic Equation Given the Roots 4 , -1/2

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