Find the Quadratic Equation - square root of 34 , square root of 34

$- \sqrt{34}$ , $\sqrt{34}$

Step 1

$x = - \sqrt{34}$ and $x = \sqrt{34}$ are the two real distinct solutions for the quadratic equation, which means that $x + \sqrt{34}$ and $x - \sqrt{34}$ are the factors of the quadratic equation.

$(x + \sqrt{34}) (x - \sqrt{34}) = 0$

Step 2

Expand $(x + \sqrt{34}) (x - \sqrt{34})$ using the FOIL Method.

Apply the distributive property.

$x (x - \sqrt{34}) + \sqrt{34} (x - \sqrt{34}) = 0$

Apply the distributive property.

$x \cdot x + x (- \sqrt{34}) + \sqrt{34} (x - \sqrt{34}) = 0$

Apply the distributive property.

$x \cdot x + x (- \sqrt{34}) + \sqrt{34} x + \sqrt{34} (- \sqrt{34}) = 0$

Step 3

Simplify terms.

Combine the opposite terms in $x \cdot x + x (- \sqrt{34}) + \sqrt{34} x + \sqrt{34} (- \sqrt{34})$ .

Reorder the factors in the terms $x (- \sqrt{34})$ and $\sqrt{34} x$ .

$x \cdot x - x \sqrt{34} + x \sqrt{34} + \sqrt{34} (- \sqrt{34}) = 0$

Add $- x \sqrt{34}$ and $x \sqrt{34}$ .

$x \cdot x + 0 + \sqrt{34} (- \sqrt{34}) = 0$

Add $x \cdot x$ and $0$ .

$x \cdot x + \sqrt{34} (- \sqrt{34}) = 0$

Simplify each term.

Multiply $x$ by $x$ .

$x^{2} + \sqrt{34} (- \sqrt{34}) = 0$

Multiply $\sqrt{34} (- \sqrt{34})$ .

Raise $\sqrt{34}$ to the power of $1$ .

$x^{2} - (\sqrt{34} \sqrt{34}) = 0$

Raise $\sqrt{34}$ to the power of $1$ .

$x^{2} - (\sqrt{34} \sqrt{34}) = 0$

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

$x^{2} - {\sqrt{34}}^{1 + 1} = 0$

Add $1$ and $1$ .

$x^{2} - {\sqrt{34}}^{2} = 0$

Rewrite ${\sqrt{34}}^{2}$ as $34$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{34}$ as $34^{\frac{1}{2}}$ .

$x^{2} - {(34^{\frac{1}{2}})}^{2} = 0$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x^{2} - 34^{\frac{1}{2} \cdot 2} = 0$

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

$x^{2} - 34^{\frac{2}{2}} = 0$

Cancel the common factor of $2$ .

Cancel the common factor.

$x^{2} - 34^{\frac{2}{2}} = 0$

Rewrite the expression.

$x^{2} - 34 = 0$

Evaluate the exponent.

$x^{2} - 1 \cdot 34 = 0$

Multiply $- 1$ by $34$ .

$x^{2} - 34 = 0$

Step 4

The standard quadratic equation using the given set of solutions ${- \sqrt{34}, \sqrt{34}}$ is $y = x^{2} - 34$ .

$y = x^{2} - 34$

Step 5

Find the Quadratic Equation - square root of 34 , square root of 34

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