Algebra Examples

$5$ , $- \frac{19}{4}$

Step 1

$x = 5$ and $x = - \frac{19}{4}$ are the two real distinct solutions for the quadratic equation, which means that $x - 5$ and $x + \frac{19}{4}$ are the factors of the quadratic equation.

$(x - 5) (x + \frac{19}{4}) = 0$

Step 2

Expand $(x - 5) (x + \frac{19}{4})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$x (x + \frac{19}{4}) - 5 (x + \frac{19}{4}) = 0$

Step 2.2

Apply the distributive property.

$x \cdot x + x (\frac{19}{4}) - 5 (x + \frac{19}{4}) = 0$

Step 2.3

Apply the distributive property.

$x \cdot x + x (\frac{19}{4}) - 5 x - 5 (\frac{19}{4}) = 0$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $x$ by $x$ .

$x^{2} + x (\frac{19}{4}) - 5 x - 5 (\frac{19}{4}) = 0$

Step 3.1.2

Combine $x$ and $\frac{19}{4}$ .

$x^{2} + \frac{x \cdot 19}{4} - 5 x - 5 (\frac{19}{4}) = 0$

Step 3.1.3

Move $19$ to the left of $x$ .

$x^{2} + \frac{19 \cdot x}{4} - 5 x - 5 (\frac{19}{4}) = 0$

Step 3.1.4

Multiply $- 5 (\frac{19}{4})$ .

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Step 3.1.4.1

Combine $- 5$ and $\frac{19}{4}$ .

$x^{2} + \frac{19 x}{4} - 5 x + \frac{- 5 \cdot 19}{4} = 0$

Step 3.1.4.2

Multiply $- 5$ by $19$ .

$x^{2} + \frac{19 x}{4} - 5 x + \frac{- 95}{4} = 0$

Step 3.1.5

Move the negative in front of the fraction.

$x^{2} + \frac{19 x}{4} - 5 x - \frac{95}{4} = 0$

Step 3.2

To write $- 5 x$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x^{2} + \frac{19 x}{4} - 5 x \cdot \frac{4}{4} - \frac{95}{4} = 0$

Step 3.3

Combine $- 5 x$ and $\frac{4}{4}$ .

$x^{2} + \frac{19 x}{4} + \frac{- 5 x \cdot 4}{4} - \frac{95}{4} = 0$

Step 3.4

Combine the numerators over the common denominator.

$x^{2} + \frac{19 x - 5 x \cdot 4}{4} - \frac{95}{4} = 0$

Step 3.5

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

$x^{2} \cdot \frac{4}{4} + \frac{19 x - 5 x \cdot 4}{4} - \frac{95}{4} = 0$

Step 3.6

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

$\frac{x^{2} \cdot 4}{4} + \frac{19 x - 5 x \cdot 4}{4} - \frac{95}{4} = 0$

Step 3.7

Combine the numerators over the common denominator.

$\frac{x^{2} \cdot 4 + 19 x - 5 x \cdot 4}{4} - \frac{95}{4} = 0$

Step 3.8

Combine the numerators over the common denominator.

$\frac{x^{2} \cdot 4 + 19 x - 5 x \cdot 4 - 95}{4} = 0$

Step 4

Simplify the numerator.

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Step 4.1

Move $4$ to the left of $x^{2}$ .

$\frac{4 \cdot x^{2} + 19 x - 5 x \cdot 4 - 95}{4} = 0$

Step 4.2

Multiply $4$ by $- 5$ .

$\frac{4 x^{2} + 19 x - 20 x - 95}{4} = 0$

Step 4.3

Subtract $20 x$ from $19 x$ .

$\frac{4 x^{2} - x - 95}{4} = 0$

Step 4.4

Factor by grouping.

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Step 4.4.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 4 \cdot - 95 = - 380$ and whose sum is $b = - 1$ .

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Step 4.4.1.1

Factor $- 1$ out of $- x$ .

$\frac{4 x^{2} - (x) - 95}{4} = 0$

Step 4.4.1.2

Rewrite $- 1$ as $19$ plus $- 20$

$\frac{4 x^{2} + (19 - 20) x - 95}{4} = 0$

Step 4.4.1.3

Apply the distributive property.

$\frac{4 x^{2} + 19 x - 20 x - 95}{4} = 0$

Step 4.4.2

Factor out the greatest common factor from each group.

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Step 4.4.2.1

Group the first two terms and the last two terms.

$\frac{(4 x^{2} + 19 x) - 20 x - 95}{4} = 0$

Step 4.4.2.2

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

$\frac{x (4 x + 19) - 5 (4 x + 19)}{4} = 0$

Step 4.4.3

Factor the polynomial by factoring out the greatest common factor, $4 x + 19$ .

$\frac{(4 x + 19) (x - 5)}{4} = 0$

Step 5

Expand $(4 x + 19) (x - 5)$ using the FOIL Method.

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Step 5.1

Apply the distributive property.

$\frac{4 x (x - 5) + 19 (x - 5)}{4} = 0$

Step 5.2

Apply the distributive property.

$\frac{4 x \cdot x + 4 x \cdot - 5 + 19 (x - 5)}{4} = 0$

Step 5.3

Apply the distributive property.

$\frac{4 x \cdot x + 4 x \cdot - 5 + 19 x + 19 \cdot - 5}{4} = 0$

Step 6

Simplify and combine like terms.

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Step 6.1

Simplify each term.

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Step 6.1.1

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

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Step 6.1.1.1

Move $x$ .

$\frac{4 (x \cdot x) + 4 x \cdot - 5 + 19 x + 19 \cdot - 5}{4} = 0$

Step 6.1.1.2

Multiply $x$ by $x$ .

$\frac{4 x^{2} + 4 x \cdot - 5 + 19 x + 19 \cdot - 5}{4} = 0$

Step 6.1.2

Multiply $- 5$ by $4$ .

$\frac{4 x^{2} - 20 x + 19 x + 19 \cdot - 5}{4} = 0$

Step 6.1.3

Multiply $19$ by $- 5$ .

$\frac{4 x^{2} - 20 x + 19 x - 95}{4} = 0$

Step 6.2

Add $- 20 x$ and $19 x$ .

$\frac{4 x^{2} - x - 95}{4} = 0$

Step 7

Split the fraction $\frac{4 x^{2} - x - 95}{4}$ into two fractions.

$\frac{4 x^{2} - x}{4} + \frac{- 95}{4} = 0$

Step 8

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

$\frac{4 x^{2}}{4} + \frac{- x}{4} + \frac{- 95}{4} = 0$

Step 9

Cancel the common factor of $4$ .

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Step 9.1

Cancel the common factor.

$\frac{4 x^{2}}{4} + \frac{- x}{4} + \frac{- 95}{4} = 0$

Step 9.2

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

$x^{2} + \frac{- x}{4} + \frac{- 95}{4} = 0$

Step 10

Move the negative in front of the fraction.

$x^{2} - \frac{x}{4} + \frac{- 95}{4} = 0$

Step 11

Move the negative in front of the fraction.

$x^{2} - \frac{x}{4} - \frac{95}{4} = 0$

Step 12

The standard quadratic equation using the given set of solutions ${5, - \frac{19}{4}}$ is $y = x^{2} - \frac{x}{4} - \frac{95}{4}$ .

$y = x^{2} - \frac{x}{4} - \frac{95}{4}$

Step 13