Find the Quadratic Equation -6/7 , 9/7

$- \frac{6}{7}$ , $\frac{9}{7}$

Step 1

$x = - \frac{6}{7}$ and $x = \frac{9}{7}$ are the two real distinct solutions for the quadratic equation, which means that $x + \frac{6}{7}$ and $x - \frac{9}{7}$ are the factors of the quadratic equation.

$(x + \frac{6}{7}) (x - \frac{9}{7}) = 0$

Step 2

Expand $(x + \frac{6}{7}) (x - \frac{9}{7})$ using the FOIL Method.

Apply the distributive property.

$x (x - \frac{9}{7}) + \frac{6}{7} \cdot (x - \frac{9}{7}) = 0$

Apply the distributive property.

$x \cdot x + x (- \frac{9}{7}) + \frac{6}{7} \cdot (x - \frac{9}{7}) = 0$

Apply the distributive property.

$x \cdot x + x (- \frac{9}{7}) + \frac{6}{7} x + \frac{6}{7} \cdot (- \frac{9}{7}) = 0$

Step 3

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x^{2} + x (- \frac{9}{7}) + \frac{6}{7} x + \frac{6}{7} \cdot (- \frac{9}{7}) = 0$

Combine $x$ and $\frac{9}{7}$ .

$x^{2} - \frac{x \cdot 9}{7} + \frac{6}{7} x + \frac{6}{7} \cdot (- \frac{9}{7}) = 0$

Move $9$ to the left of $x$ .

$x^{2} - \frac{9 \cdot x}{7} + \frac{6}{7} x + \frac{6}{7} \cdot (- \frac{9}{7}) = 0$

Combine $\frac{6}{7}$ and $x$ .

$x^{2} - \frac{9 x}{7} + \frac{6 x}{7} + \frac{6}{7} \cdot (- \frac{9}{7}) = 0$

Multiply $\frac{6}{7} (- \frac{9}{7})$ .

Multiply $\frac{6}{7}$ by $\frac{9}{7}$ .

$x^{2} - \frac{9 x}{7} + \frac{6 x}{7} - \frac{6 \cdot 9}{7 \cdot 7} = 0$

Multiply $6$ by $9$ .

$x^{2} - \frac{9 x}{7} + \frac{6 x}{7} - \frac{54}{7 \cdot 7} = 0$

Multiply $7$ by $7$ .

$x^{2} - \frac{9 x}{7} + \frac{6 x}{7} - \frac{54}{49} = 0$

Combine the numerators over the common denominator.

$x^{2} + \frac{- 9 x + 6 x}{7} - \frac{54}{49} = 0$

Step 4

Add $- 9 x$ and $6 x$ .

$x^{2} + \frac{- 3 x}{7} - \frac{54}{49} = 0$

Step 5

Move the negative in front of the fraction.

$x^{2} - \frac{3 x}{7} - \frac{54}{49} = 0$

Step 6

The standard quadratic equation using the given set of solutions ${- \frac{6}{7}, \frac{9}{7}}$ is $y = x^{2} - \frac{3 x}{7} - \frac{54}{49}$ .

$y = x^{2} - \frac{3 x}{7} - \frac{54}{49}$

Step 7

Find the Quadratic Equation -6/7 , 9/7

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