Solve for x square root of x=2-x^2

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

Step 1

To remove the radical on the left side of the equation, square both sides of the equation.

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

Step 2

Simplify each side of the equation.

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

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

Simplify the left side.

Simplify ${(x^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${(x^{\frac{1}{2}})}^{2}$ .

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$x^{1} = {(2 - x^{2})}^{2}$

Simplify.

$x = {(2 - x^{2})}^{2}$

Simplify the right side.

Simplify ${(2 - x^{2})}^{2}$ .

Rewrite ${(2 - x^{2})}^{2}$ as $(2 - x^{2}) (2 - x^{2})$ .

$x = (2 - x^{2}) (2 - x^{2})$

Expand $(2 - x^{2}) (2 - x^{2})$ using the FOIL Method.

Apply the distributive property.

$x = 2 (2 - x^{2}) - x^{2} (2 - x^{2})$

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $2$ by $2$ .

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

Multiply $- 1$ by $2$ .

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

Multiply $2$ by $- 1$ .

$x = 4 - 2 x^{2} - 2 x^{2} - x^{2} (- x^{2})$

Rewrite using the commutative property of multiplication.

$x = 4 - 2 x^{2} - 2 x^{2} - 1 \cdot - 1 x^{2} x^{2}$

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

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

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

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

Add $2$ and $2$ .

$x = 4 - 2 x^{2} - 2 x^{2} - 1 \cdot - 1 x^{4}$

Multiply $- 1$ by $- 1$ .

$x = 4 - 2 x^{2} - 2 x^{2} + 1 x^{4}$

Multiply $x^{4}$ by $1$ .

$x = 4 - 2 x^{2} - 2 x^{2} + x^{4}$

Subtract $2 x^{2}$ from $- 2 x^{2}$ .

$x = 4 - 4 x^{2} + x^{4}$

Step 3

Solve for $x$ .

Reorder the polynomial.

$x^{4} - 4 x^{2} - x + 4$

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1, \pm 2, \pm 4$

$q = \pm 1$

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 2, \pm 4$

Substitute the possible roots one by one into the polynomial to find the actual roots. Simplify to check if the value is $0$ , which means it is a root.

$4 - 4 {(1)}^{2} + {(1)}^{4} - (1)$

Simplify the expression. In this case, the expression is equal to $0$ so $x = 1$ is a root of the polynomial.

Simplify each term.

One to any power is one.

$4 - 4 \cdot 1 + {(1)}^{4} - (1)$

Multiply $- 4$ by $1$ .

$4 - 4 + {(1)}^{4} - (1)$

One to any power is one.

$4 - 4 + 1 - (1)$

Multiply $- 1$ by $1$ .

$4 - 4 + 1 - 1$

Simplify by adding and subtracting.

Subtract $4$ from $4$ .

$0 + 1 - 1$

Add $0$ and $1$ .

$1 - 1$

Subtract $1$ from $1$ .

$0$

Since $1$ is a known root, divide the polynomial by $x - 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{4 - 4 x^{2} + x^{4} - x}{x - 1}$

Next, find the roots of the remaining polynomial. The order of the polynomial has been reduced by $1$ .

Place the numbers representing the divisor and the dividend into a division-like configuration.

$1$	$1$	$0$	$- 4$	$- 1$	$4$
	\xa0				\xa0

The first number in the dividend $(1)$ is put into the first position of the result area (below the horizontal line).

$1$	$1$	$0$	$- 4$	$- 1$	$4$
	\xa0				\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(1)$ and place the result of $(1)$ under the next term in the dividend $(0)$ .

$1$	$1$	$0$	$- 4$	$- 1$	$4$
	\xa0	$1$			\xa0
	$1$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$1$	$1$	$0$	$- 4$	$- 1$	$4$
	\xa0	$1$			\xa0
	$1$	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(1)$ and place the result of $(1)$ under the next term in the dividend $(- 4)$ .

$1$	$1$	$0$	$- 4$	$- 1$	$4$
	\xa0	$1$	$1$		\xa0
	$1$	$1$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$1$	$1$	$0$	$- 4$	$- 1$	$4$
	\xa0	$1$	$1$		\xa0
	$1$	$1$	$- 3$

Multiply the newest entry in the result $(- 3)$ by the divisor $(1)$ and place the result of $(- 3)$ under the next term in the dividend $(- 1)$ .

$1$	$1$	$0$	$- 4$	$- 1$	$4$
	\xa0	$1$	$1$	$- 3$	\xa0
	$1$	$1$	$- 3$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$1$	$1$	$0$	$- 4$	$- 1$	$4$
	\xa0	$1$	$1$	$- 3$	\xa0
	$1$	$1$	$- 3$	$- 4$

Multiply the newest entry in the result $(- 4)$ by the divisor $(1)$ and place the result of $(- 4)$ under the next term in the dividend $(4)$ .

$1$	$1$	$0$	$- 4$	$- 1$	$4$
	\xa0	$1$	$1$	$- 3$	$- 4$
	$1$	$1$	$- 3$	$- 4$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$1$	$1$	$0$	$- 4$	$- 1$	$4$
	\xa0	$1$	$1$	$- 3$	$- 4$
	$1$	$1$	$- 3$	$- 4$	$0$

All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.

$1 x^{3} + 1 x^{2} + (- 3) x - 4$

Simplify the quotient polynomial.

$x^{3} + x^{2} - 3 x - 4$

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x \approx 1.8311772$

The polynomial can be written as a set of linear factors.

$(x - 1) (x - 1.8311772)$

These are the roots (zeros) of the polynomial $x^{4} - 4 x^{2} - x + 4$ .

$x = 1, 1.8311772$

Step 4

Exclude the solutions that do not make $\sqrt{x} = 2 - x^{2}$ true.

$x = 1$

Solve for x square root of x=2-x^2

Related Questions

Support LCMGCF.com by disabling your adblocker.