Algebra Examples

${- \frac{4}{5}, \frac{5}{7}}$

Step 1

$x = - \frac{4}{5}$ and $x = \frac{5}{7}$ are the two real distinct solutions for the quadratic equation, which means that $x + \frac{4}{5}$ and $x - \frac{5}{7}$ are the factors of the quadratic equation.

$(x + \frac{4}{5}) (x - \frac{5}{7}) = 0$

Step 2

Expand $(x + \frac{4}{5}) (x - \frac{5}{7})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$x (x - \frac{5}{7}) + \frac{4}{5} \cdot (x - \frac{5}{7}) = 0$

Step 2.2

Apply the distributive property.

$x \cdot x + x (- \frac{5}{7}) + \frac{4}{5} \cdot (x - \frac{5}{7}) = 0$

Step 2.3

Apply the distributive property.

$x \cdot x + x (- \frac{5}{7}) + \frac{4}{5} x + \frac{4}{5} \cdot (- \frac{5}{7}) = 0$

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 $x$ .

$x^{2} + x (- \frac{5}{7}) + \frac{4}{5} x + \frac{4}{5} \cdot (- \frac{5}{7}) = 0$

Step 3.1.2

Combine $x$ and $\frac{5}{7}$ .

$x^{2} - \frac{x \cdot 5}{7} + \frac{4}{5} x + \frac{4}{5} \cdot (- \frac{5}{7}) = 0$

Step 3.1.3

Move $5$ to the left of $x$ .

$x^{2} - \frac{5 \cdot x}{7} + \frac{4}{5} x + \frac{4}{5} \cdot (- \frac{5}{7}) = 0$

Step 3.1.4

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

$x^{2} - \frac{5 x}{7} + \frac{4 x}{5} + \frac{4}{5} \cdot (- \frac{5}{7}) = 0$

Step 3.1.5

Cancel the common factor of $5$ .

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Step 3.1.5.1

Move the leading negative in $- \frac{5}{7}$ into the numerator.

$x^{2} - \frac{5 x}{7} + \frac{4 x}{5} + \frac{4}{5} \cdot \frac{- 5}{7} = 0$

Step 3.1.5.2

Factor $5$ out of $- 5$ .

$x^{2} - \frac{5 x}{7} + \frac{4 x}{5} + \frac{4}{5} \cdot \frac{5 (- 1)}{7} = 0$

Step 3.1.5.3

Cancel the common factor.

$x^{2} - \frac{5 x}{7} + \frac{4 x}{5} + \frac{4}{5} \cdot \frac{5 \cdot - 1}{7} = 0$

Step 3.1.5.4

Rewrite the expression.

$x^{2} - \frac{5 x}{7} + \frac{4 x}{5} + 4 (\frac{- 1}{7}) = 0$

Step 3.1.6

Combine $4$ and $\frac{- 1}{7}$ .

$x^{2} - \frac{5 x}{7} + \frac{4 x}{5} + \frac{4 \cdot - 1}{7} = 0$

Step 3.1.7

Multiply $4$ by $- 1$ .

$x^{2} - \frac{5 x}{7} + \frac{4 x}{5} + \frac{- 4}{7} = 0$

Step 3.1.8

Move the negative in front of the fraction.

$x^{2} - \frac{5 x}{7} + \frac{4 x}{5} - \frac{4}{7} = 0$

Step 3.2

To write $- \frac{5 x}{7}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$x^{2} - \frac{5 x}{7} \cdot \frac{5}{5} + \frac{4 x}{5} - \frac{4}{7} = 0$

Step 3.3

To write $\frac{4 x}{5}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$x^{2} - \frac{5 x}{7} \cdot \frac{5}{5} + \frac{4 x}{5} \cdot \frac{7}{7} - \frac{4}{7} = 0$

Step 3.4

Write each expression with a common denominator of $35$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.4.1

Multiply $\frac{5 x}{7}$ by $\frac{5}{5}$ .

$x^{2} - \frac{5 x \cdot 5}{7 \cdot 5} + \frac{4 x}{5} \cdot \frac{7}{7} - \frac{4}{7} = 0$

Step 3.4.2

Multiply $7$ by $5$ .

$x^{2} - \frac{5 x \cdot 5}{35} + \frac{4 x}{5} \cdot \frac{7}{7} - \frac{4}{7} = 0$

Step 3.4.3

Multiply $\frac{4 x}{5}$ by $\frac{7}{7}$ .

$x^{2} - \frac{5 x \cdot 5}{35} + \frac{4 x \cdot 7}{5 \cdot 7} - \frac{4}{7} = 0$

Step 3.4.4

Multiply $5$ by $7$ .

$x^{2} - \frac{5 x \cdot 5}{35} + \frac{4 x \cdot 7}{35} - \frac{4}{7} = 0$

Step 3.5

Combine the numerators over the common denominator.

$x^{2} + \frac{- 5 x \cdot 5 + 4 x \cdot 7}{35} - \frac{4}{7} = 0$

Step 3.6

To write $x^{2}$ as a fraction with a common denominator, multiply by $\frac{35}{35}$ .

$x^{2} \cdot \frac{35}{35} + \frac{- 5 x \cdot 5 + 4 x \cdot 7}{35} - \frac{4}{7} = 0$

Step 3.7

Combine $x^{2}$ and $\frac{35}{35}$ .

$\frac{x^{2} \cdot 35}{35} + \frac{- 5 x \cdot 5 + 4 x \cdot 7}{35} - \frac{4}{7} = 0$

Step 3.8

Combine the numerators over the common denominator.

$\frac{x^{2} \cdot 35 - 5 x \cdot 5 + 4 x \cdot 7}{35} - \frac{4}{7} = 0$

Step 3.9

To write $- \frac{4}{7}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{x^{2} \cdot 35 - 5 x \cdot 5 + 4 x \cdot 7}{35} - \frac{4}{7} \cdot \frac{5}{5} = 0$

Step 3.10

Write each expression with a common denominator of $35$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.10.1

Multiply $\frac{4}{7}$ by $\frac{5}{5}$ .

$\frac{x^{2} \cdot 35 - 5 x \cdot 5 + 4 x \cdot 7}{35} - \frac{4 \cdot 5}{7 \cdot 5} = 0$

Step 3.10.2

Multiply $7$ by $5$ .

$\frac{x^{2} \cdot 35 - 5 x \cdot 5 + 4 x \cdot 7}{35} - \frac{4 \cdot 5}{35} = 0$

Step 3.11

Combine the numerators over the common denominator.

$\frac{x^{2} \cdot 35 - 5 x \cdot 5 + 4 x \cdot 7 - 4 \cdot 5}{35} = 0$

Step 4

Simplify the numerator.

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Step 4.1

Move $35$ to the left of $x^{2}$ .

$\frac{35 \cdot x^{2} - 5 x \cdot 5 + 4 x \cdot 7 - 4 \cdot 5}{35} = 0$

Step 4.2

Multiply $5$ by $- 5$ .

$\frac{35 x^{2} - 25 x + 4 x \cdot 7 - 4 \cdot 5}{35} = 0$

Step 4.3

Multiply $7$ by $4$ .

$\frac{35 x^{2} - 25 x + 28 x - 4 \cdot 5}{35} = 0$

Step 4.4

Multiply $- 4$ by $5$ .

$\frac{35 x^{2} - 25 x + 28 x - 20}{35} = 0$

Step 4.5

Add $- 25 x$ and $28 x$ .

$\frac{35 x^{2} + 3 x - 20}{35} = 0$

Step 4.6

Factor by grouping.

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Step 4.6.1

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 $a \cdot c = 35 \cdot - 20 = - 700$ and whose sum is $b = 3$ .

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Step 4.6.1.1

Factor $3$ out of $3 x$ .

$\frac{35 x^{2} + 3 (x) - 20}{35} = 0$

Step 4.6.1.2

Rewrite $3$ as $- 25$ plus $28$

$\frac{35 x^{2} + (- 25 + 28) x - 20}{35} = 0$

Step 4.6.1.3

Apply the distributive property.

$\frac{35 x^{2} - 25 x + 28 x - 20}{35} = 0$

Step 4.6.2

Factor out the greatest common factor from each group.

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Step 4.6.2.1

Group the first two terms and the last two terms.

$\frac{(35 x^{2} - 25 x) + 28 x - 20}{35} = 0$

Step 4.6.2.2

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

$\frac{5 x (7 x - 5) + 4 (7 x - 5)}{35} = 0$

Step 4.6.3

Factor the polynomial by factoring out the greatest common factor, $7 x - 5$ .

$\frac{(7 x - 5) (5 x + 4)}{35} = 0$

Step 5

Expand $(7 x - 5) (5 x + 4)$ using the FOIL Method.

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Step 5.1

Apply the distributive property.

$\frac{7 x (5 x + 4) - 5 (5 x + 4)}{35} = 0$

Step 5.2

Apply the distributive property.

$\frac{7 x (5 x) + 7 x \cdot 4 - 5 (5 x + 4)}{35} = 0$

Step 5.3

Apply the distributive property.

$\frac{7 x (5 x) + 7 x \cdot 4 - 5 (5 x) - 5 \cdot 4}{35} = 0$

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

Rewrite using the commutative property of multiplication.

$\frac{7 \cdot (5 x \cdot x) + 7 x \cdot 4 - 5 (5 x) - 5 \cdot 4}{35} = 0$

Step 6.1.2

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

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Step 6.1.2.1

Move $x$ .

$\frac{7 \cdot (5 (x \cdot x)) + 7 x \cdot 4 - 5 (5 x) - 5 \cdot 4}{35} = 0$

Step 6.1.2.2

Multiply $x$ by $x$ .

$\frac{7 \cdot (5 x^{2}) + 7 x \cdot 4 - 5 (5 x) - 5 \cdot 4}{35} = 0$

Step 6.1.3

Multiply $7$ by $5$ .

$\frac{35 x^{2} + 7 x \cdot 4 - 5 (5 x) - 5 \cdot 4}{35} = 0$

Step 6.1.4

Multiply $4$ by $7$ .

$\frac{35 x^{2} + 28 x - 5 (5 x) - 5 \cdot 4}{35} = 0$

Step 6.1.5

Multiply $5$ by $- 5$ .

$\frac{35 x^{2} + 28 x - 25 x - 5 \cdot 4}{35} = 0$

Step 6.1.6

Multiply $- 5$ by $4$ .

$\frac{35 x^{2} + 28 x - 25 x - 20}{35} = 0$

Step 6.2

Subtract $25 x$ from $28 x$ .

$\frac{35 x^{2} + 3 x - 20}{35} = 0$

Step 7

Split the fraction $\frac{35 x^{2} + 3 x - 20}{35}$ into two fractions.

$\frac{35 x^{2} + 3 x}{35} + \frac{- 20}{35} = 0$

Step 8

Split the fraction $\frac{35 x^{2} + 3 x}{35}$ into two fractions.

$\frac{35 x^{2}}{35} + \frac{3 x}{35} + \frac{- 20}{35} = 0$

Step 9

Cancel the common factor of $35$ .

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Step 9.1

Cancel the common factor.

$\frac{35 x^{2}}{35} + \frac{3 x}{35} + \frac{- 20}{35} = 0$

Step 9.2

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

$x^{2} + \frac{3 x}{35} + \frac{- 20}{35} = 0$

Step 10

Cancel the common factor of $- 20$ and $35$ .

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Step 10.1

Factor $5$ out of $- 20$ .

$x^{2} + \frac{3 x}{35} + \frac{5 (- 4)}{35} = 0$

Step 10.2

Cancel the common factors.

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Step 10.2.1

Factor $5$ out of $35$ .

$x^{2} + \frac{3 x}{35} + \frac{5 \cdot - 4}{5 \cdot 7} = 0$

Step 10.2.2

Cancel the common factor.

$x^{2} + \frac{3 x}{35} + \frac{5 \cdot - 4}{5 \cdot 7} = 0$

Step 10.2.3

Rewrite the expression.

$x^{2} + \frac{3 x}{35} + \frac{- 4}{7} = 0$

Step 11

Move the negative in front of the fraction.

$x^{2} + \frac{3 x}{35} - \frac{4}{7} = 0$

Step 12

The standard quadratic equation using the given set of solutions ${- \frac{4}{5}, \frac{5}{7}}$ is $y = x^{2} + \frac{3 x}{35} - \frac{4}{7}$ .

$y = x^{2} + \frac{3 x}{35} - \frac{4}{7}$

Step 13