Find All Complex Solutions 8x^2-3 = square root of 16x+9

$8 x^{2} - 3 = \sqrt{16 x + 9}$

Step 1

Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.

$\sqrt{16 x + 9} = 8 x^{2} - 3$

Step 2

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

${\sqrt{16 x + 9}}^{2} = {(8 x^{2} - 3)}^{2}$

Step 3

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{16 x + 9}$ as ${(16 x + 9)}^{\frac{1}{2}}$ .

${({(16 x + 9)}^{\frac{1}{2}})}^{2} = {(8 x^{2} - 3)}^{2}$

Simplify the left side.

Simplify ${({(16 x + 9)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(16 x + 9)}^{\frac{1}{2}})}^{2}$ .

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

${(16 x + 9)}^{\frac{1}{2} \cdot 2} = {(8 x^{2} - 3)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(16 x + 9)}^{\frac{1}{2} \cdot 2} = {(8 x^{2} - 3)}^{2}$

Rewrite the expression.

${(16 x + 9)}^{1} = {(8 x^{2} - 3)}^{2}$

Simplify.

$16 x + 9 = {(8 x^{2} - 3)}^{2}$

Simplify the right side.

Simplify ${(8 x^{2} - 3)}^{2}$ .

Rewrite ${(8 x^{2} - 3)}^{2}$ as $(8 x^{2} - 3) (8 x^{2} - 3)$ .

$16 x + 9 = (8 x^{2} - 3) (8 x^{2} - 3)$

Expand $(8 x^{2} - 3) (8 x^{2} - 3)$ using the FOIL Method.

Apply the distributive property.

$16 x + 9 = 8 x^{2} (8 x^{2} - 3) - 3 (8 x^{2} - 3)$

Apply the distributive property.

$16 x + 9 = 8 x^{2} (8 x^{2}) + 8 x^{2} \cdot - 3 - 3 (8 x^{2} - 3)$

Apply the distributive property.

$16 x + 9 = 8 x^{2} (8 x^{2}) + 8 x^{2} \cdot - 3 - 3 (8 x^{2}) - 3 \cdot - 3$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$16 x + 9 = 8 \cdot 8 x^{2} x^{2} + 8 x^{2} \cdot - 3 - 3 (8 x^{2}) - 3 \cdot - 3$

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

Move $x^{2}$ .

$16 x + 9 = 8 \cdot 8 (x^{2} x^{2}) + 8 x^{2} \cdot - 3 - 3 (8 x^{2}) - 3 \cdot - 3$

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

$16 x + 9 = 8 \cdot 8 x^{2 + 2} + 8 x^{2} \cdot - 3 - 3 (8 x^{2}) - 3 \cdot - 3$

Add $2$ and $2$ .

$16 x + 9 = 8 \cdot 8 x^{4} + 8 x^{2} \cdot - 3 - 3 (8 x^{2}) - 3 \cdot - 3$

Multiply $8$ by $8$ .

$16 x + 9 = 64 x^{4} + 8 x^{2} \cdot - 3 - 3 (8 x^{2}) - 3 \cdot - 3$

Multiply $- 3$ by $8$ .

$16 x + 9 = 64 x^{4} - 24 x^{2} - 3 (8 x^{2}) - 3 \cdot - 3$

Multiply $8$ by $- 3$ .

$16 x + 9 = 64 x^{4} - 24 x^{2} - 24 x^{2} - 3 \cdot - 3$

Multiply $- 3$ by $- 3$ .

$16 x + 9 = 64 x^{4} - 24 x^{2} - 24 x^{2} + 9$

Subtract $24 x^{2}$ from $- 24 x^{2}$ .

$16 x + 9 = 64 x^{4} - 48 x^{2} + 9$

Step 4

Solve for $x$ .

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$64 x^{4} - 48 x^{2} + 9 = 16 x + 9$

Subtract $16 x$ from both sides of the equation.

$64 x^{4} - 48 x^{2} + 9 - 16 x = 9$

Subtract $9$ from both sides of the equation.

$64 x^{4} - 48 x^{2} + 9 - 16 x - 9 = 0$

Combine the opposite terms in $64 x^{4} - 48 x^{2} + 9 - 16 x - 9$ .

Subtract $9$ from $9$ .

$64 x^{4} - 48 x^{2} - 16 x + 0 = 0$

Add $64 x^{4} - 48 x^{2} - 16 x$ and $0$ .

$64 x^{4} - 48 x^{2} - 16 x = 0$

Factor the left side of the equation.

Factor $16 x$ out of $64 x^{4} - 48 x^{2} - 16 x$ .

Factor $16 x$ out of $64 x^{4}$ .

$16 x (4 x^{3}) - 48 x^{2} - 16 x = 0$

Factor $16 x$ out of $- 48 x^{2}$ .

$16 x (4 x^{3}) + 16 x (- 3 x) - 16 x = 0$

Factor $16 x$ out of $- 16 x$ .

$16 x (4 x^{3}) + 16 x (- 3 x) + 16 x (- 1) = 0$

Factor $16 x$ out of $16 x (4 x^{3}) + 16 x (- 3 x)$ .

$16 x (4 x^{3} - 3 x) + 16 x (- 1) = 0$

Factor $16 x$ out of $16 x (4 x^{3} - 3 x) + 16 x (- 1)$ .

$16 x (4 x^{3} - 3 x - 1) = 0$

Factor $4 x^{3} - 3 x - 1$ using the rational roots test.

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.

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

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

$\pm 1, \pm 0.25, \pm 0.5$

Substitute $- 0.5$ and simplify the expression. In this case, the expression is equal to $0$ so $- 0.5$ is a root of the polynomial.

Substitute $- 0.5$ into the polynomial.

$4 {(- 0.5)}^{3} - 3 \cdot - 0.5 - 1$

Raise $- 0.5$ to the power of $3$ .

$4 \cdot - 0.125 - 3 \cdot - 0.5 - 1$

Multiply $4$ by $- 0.125$ .

$- 0.5 - 3 \cdot - 0.5 - 1$

Multiply $- 3$ by $- 0.5$ .

$- 0.5 + 1.5 - 1$

Add $- 0.5$ and $1.5$ .

$1 - 1$

Subtract $1$ from $1$ .

$0$

Since $- 0.5$ is a known root, divide the polynomial by $2 x + 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{4 x^{3} - 3 x - 1}{2 x + 1}$

Divide $4 x^{3} - 3 x - 1$ by $2 x + 1$ .

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$2 x$

$1$

$4 x^{3}$

$0 x^{2}$

$3 x$

$1$

Divide the highest order term in the dividend $4 x^{3}$ by the highest order term in divisor $2 x$ .

					$2 x^{2}$
	$2 x$	+	$1$		$4 x^{3}$	+	$0 x^{2}$	-	$3 x$	-	$1$

Multiply the new quotient term by the divisor.

				$2 x^{2}$
$2 x$	+	$1$		$4 x^{3}$	+	$0 x^{2}$	-	$3 x$	-	$1$
			+	$4 x^{3}$	+	$2 x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $4 x^{3} + 2 x^{2}$

				$2 x^{2}$
$2 x$	+	$1$		$4 x^{3}$	+	$0 x^{2}$	-	$3 x$	-	$1$
			-	$4 x^{3}$	-	$2 x^{2}$

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$2 x^{2}$
$2 x$	+	$1$		$4 x^{3}$	+	$0 x^{2}$	-	$3 x$	-	$1$
			-	$4 x^{3}$	-	$2 x^{2}$
					-	$2 x^{2}$

Pull the next terms from the original dividend down into the current dividend.

				$2 x^{2}$
$2 x$	+	$1$		$4 x^{3}$	+	$0 x^{2}$	-	$3 x$	-	$1$
			-	$4 x^{3}$	-	$2 x^{2}$
					-	$2 x^{2}$	-	$3 x$

Divide the highest order term in the dividend $- 2 x^{2}$ by the highest order term in divisor $2 x$ .

				$2 x^{2}$	-	$x$
$2 x$	+	$1$		$4 x^{3}$	+	$0 x^{2}$	-	$3 x$	-	$1$
			-	$4 x^{3}$	-	$2 x^{2}$
					-	$2 x^{2}$	-	$3 x$

Multiply the new quotient term by the divisor.

				$2 x^{2}$	-	$x$
$2 x$	+	$1$		$4 x^{3}$	+	$0 x^{2}$	-	$3 x$	-	$1$
			-	$4 x^{3}$	-	$2 x^{2}$
					-	$2 x^{2}$	-	$3 x$
					-	$2 x^{2}$	-	$x$

The expression needs to be subtracted from the dividend, so change all the signs in $- 2 x^{2} - x$

				$2 x^{2}$	-	$x$
$2 x$	+	$1$		$4 x^{3}$	+	$0 x^{2}$	-	$3 x$	-	$1$
			-	$4 x^{3}$	-	$2 x^{2}$
					-	$2 x^{2}$	-	$3 x$
					+	$2 x^{2}$	+	$x$

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$2 x^{2}$	-	$x$
$2 x$	+	$1$		$4 x^{3}$	+	$0 x^{2}$	-	$3 x$	-	$1$
			-	$4 x^{3}$	-	$2 x^{2}$
					-	$2 x^{2}$	-	$3 x$
					+	$2 x^{2}$	+	$x$
							-	$2 x$

Pull the next terms from the original dividend down into the current dividend.

				$2 x^{2}$	-	$x$
$2 x$	+	$1$		$4 x^{3}$	+	$0 x^{2}$	-	$3 x$	-	$1$
			-	$4 x^{3}$	-	$2 x^{2}$
					-	$2 x^{2}$	-	$3 x$
					+	$2 x^{2}$	+	$x$
							-	$2 x$	-	$1$

Divide the highest order term in the dividend $- 2 x$ by the highest order term in divisor $2 x$ .

				$2 x^{2}$	-	$x$	-	$1$
$2 x$	+	$1$		$4 x^{3}$	+	$0 x^{2}$	-	$3 x$	-	$1$
			-	$4 x^{3}$	-	$2 x^{2}$
					-	$2 x^{2}$	-	$3 x$
					+	$2 x^{2}$	+	$x$
							-	$2 x$	-	$1$

Multiply the new quotient term by the divisor.

				$2 x^{2}$	-	$x$	-	$1$
$2 x$	+	$1$		$4 x^{3}$	+	$0 x^{2}$	-	$3 x$	-	$1$
			-	$4 x^{3}$	-	$2 x^{2}$
					-	$2 x^{2}$	-	$3 x$
					+	$2 x^{2}$	+	$x$
							-	$2 x$	-	$1$
							-	$2 x$	-	$1$

The expression needs to be subtracted from the dividend, so change all the signs in $- 2 x - 1$

				$2 x^{2}$	-	$x$	-	$1$
$2 x$	+	$1$		$4 x^{3}$	+	$0 x^{2}$	-	$3 x$	-	$1$
			-	$4 x^{3}$	-	$2 x^{2}$
					-	$2 x^{2}$	-	$3 x$
					+	$2 x^{2}$	+	$x$
							-	$2 x$	-	$1$
							+	$2 x$	+	$1$

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$2 x^{2}$	-	$x$	-	$1$
$2 x$	+	$1$		$4 x^{3}$	+	$0 x^{2}$	-	$3 x$	-	$1$
			-	$4 x^{3}$	-	$2 x^{2}$
					-	$2 x^{2}$	-	$3 x$
					+	$2 x^{2}$	+	$x$
							-	$2 x$	-	$1$
							+	$2 x$	+	$1$
										$0$

Since the remander is $0$ , the final answer is the quotient.

$2 x^{2} - x - 1$

Write $4 x^{3} - 3 x - 1$ as a set of factors.

$16 x ((2 x + 1) (2 x^{2} - x - 1)) = 0$

Factor by grouping.

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 $a \cdot c = 2 \cdot - 1 = - 2$ and whose sum is $b = - 1$ .

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

$16 x ((2 x + 1) (2 x^{2} - (x) - 1)) = 0$

Rewrite $- 1$ as $1$ plus $- 2$

$16 x ((2 x + 1) (2 x^{2} + (1 - 2) x - 1)) = 0$

Apply the distributive property.

$16 x ((2 x + 1) (2 x^{2} + 1 x - 2 x - 1)) = 0$

Multiply $x$ by $1$ .

$16 x ((2 x + 1) (2 x^{2} + x - 2 x - 1)) = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$16 x ((2 x + 1) ((2 x^{2} + x) - 2 x - 1)) = 0$

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

$16 x ((2 x + 1) (x (2 x + 1) - (2 x + 1))) = 0$

Factor the polynomial by factoring out the greatest common factor, $2 x + 1$ .

$16 x ((2 x + 1) ((2 x + 1) (x - 1))) = 0$

Remove unnecessary parentheses.

$16 x ((2 x + 1) (2 x + 1) (x - 1)) = 0$

Factor.

Combine like factors.

Raise $2 x + 1$ to the power of $1$ .

$16 x ((2 x + 1) (2 x + 1) (x - 1)) = 0$

Raise $2 x + 1$ to the power of $1$ .

$16 x ((2 x + 1) (2 x + 1) (x - 1)) = 0$

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

$16 x ({(2 x + 1)}^{1 + 1} (x - 1)) = 0$

Add $1$ and $1$ .

$16 x ({(2 x + 1)}^{2} (x - 1)) = 0$

Remove unnecessary parentheses.

$16 x {(2 x + 1)}^{2} (x - 1) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

${(2 x + 1)}^{2} = 0$

$x - 1 = 0$

Set $x$ equal to $0$ .

$x = 0$

Set ${(2 x + 1)}^{2}$ equal to $0$ and solve for $x$ .

Set ${(2 x + 1)}^{2}$ equal to $0$ .

${(2 x + 1)}^{2} = 0$

Solve ${(2 x + 1)}^{2} = 0$ for $x$ .

Set the $2 x + 1$ equal to $0$ .

$2 x + 1 = 0$

Solve for $x$ .

Subtract $1$ from both sides of the equation.

$2 x = - 1$

Divide each term in $2 x = - 1$ by $2$ and simplify.

Divide each term in $2 x = - 1$ by $2$ .

$\frac{2 x}{2} = \frac{- 1}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 1}{2}$

Divide $x$ by $1$ .

$x = \frac{- 1}{2}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{1}{2}$

Set $x - 1$ equal to $0$ and solve for $x$ .

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

The final solution is all the values that make $16 x {(2 x + 1)}^{2} (x - 1) = 0$ true.

$x = 0, - \frac{1}{2}, 1$

Step 5

Exclude the solutions that do not make $8 x^{2} - 3 = \sqrt{16 x + 9}$ true.

$x = 1$

Find All Complex Solutions 8x^2-3 = square root of 16x+9

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