Solve for x |3x-4|=2.5 square root of 3-x

$| 3 x - 4 | = 2.5 \sqrt{3 - x}$

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.

$2.5 \sqrt{3 - x} = | 3 x - 4 |$

Step 2

Rewrite the equation as $| 3 x - 4 | = 2.5 \sqrt{3 - x}$ .

$| 3 x - 4 | = 2.5 \sqrt{3 - x}$

Step 3

To remove the absolute value on the left side of the equation, use the property $| a | = \sqrt{a^{2}}$ .

$\sqrt{{(3 x - 4)}^{2}} = 2.5 \sqrt{3 - x}$

Step 4

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

${\sqrt{{(3 x - 4)}^{2}}}^{2} = {(2.5 \sqrt{3 - x})}^{2}$

Step 5

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{{(3 x - 4)}^{2}}$ as ${(3 x - 4)}^{\frac{2}{2}}$ .

${({(3 x - 4)}^{\frac{2}{2}})}^{2} = {(2.5 \sqrt{3 - x})}^{2}$

Divide $2$ by $2$ .

${({(3 x - 4)}^{1})}^{2} = {(2.5 \sqrt{3 - x})}^{2}$

Simplify the left side.

Multiply the exponents in ${({(3 x - 4)}^{1})}^{2}$ .

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

${(3 x - 4)}^{1 \cdot 2} = {(2.5 \sqrt{3 - x})}^{2}$

Multiply $2$ by $1$ .

${(3 x - 4)}^{2} = {(2.5 \sqrt{3 - x})}^{2}$

Simplify the right side.

Simplify ${(2.5 \sqrt{3 - x})}^{2}$ .

Simplify the expression.

Apply the product rule to $2.5 \sqrt{3 - x}$ .

${(3 x - 4)}^{2} = {2.5}^{2} {\sqrt{3 - x}}^{2}$

Raise $2.5$ to the power of $2$ .

${(3 x - 4)}^{2} = 6.25 {\sqrt{3 - x}}^{2}$

Rewrite ${\sqrt{3 - x}}^{2}$ as $3 - x$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 - x}$ as ${(3 - x)}^{\frac{1}{2}}$ .

${(3 x - 4)}^{2} = 6.25 {({(3 - x)}^{\frac{1}{2}})}^{2}$

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

${(3 x - 4)}^{2} = 6.25 {(3 - x)}^{\frac{1}{2} \cdot 2}$

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

${(3 x - 4)}^{2} = 6.25 {(3 - x)}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(3 x - 4)}^{2} = 6.25 {(3 - x)}^{\frac{2}{2}}$

Rewrite the expression.

${(3 x - 4)}^{2} = 6.25 {(3 - x)}^{1}$

Simplify.

${(3 x - 4)}^{2} = 6.25 (3 - x)$

Apply the distributive property.

${(3 x - 4)}^{2} = 6.25 \cdot 3 + 6.25 (- x)$

Multiply.

Multiply $6.25$ by $3$ .

${(3 x - 4)}^{2} = 18.75 + 6.25 (- x)$

Multiply $- 1$ by $6.25$ .

${(3 x - 4)}^{2} = 18.75 - 6.25 x$

Step 6

Solve for $x$ .

Move all terms containing $x$ to the left side of the equation.

Add $6.25 x$ to both sides of the equation.

${(3 x - 4)}^{2} + 6.25 x = 18.75$

Simplify each term.

Rewrite ${(3 x - 4)}^{2}$ as $(3 x - 4) (3 x - 4)$ .

$(3 x - 4) (3 x - 4) + 6.25 x = 18.75$

Expand $(3 x - 4) (3 x - 4)$ using the FOIL Method.

Apply the distributive property.

$3 x (3 x - 4) - 4 (3 x - 4) + 6.25 x = 18.75$

Apply the distributive property.

$3 x (3 x) + 3 x \cdot - 4 - 4 (3 x - 4) + 6.25 x = 18.75$

Apply the distributive property.

$3 x (3 x) + 3 x \cdot - 4 - 4 (3 x) - 4 \cdot - 4 + 6.25 x = 18.75$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$3 \cdot 3 x \cdot x + 3 x \cdot - 4 - 4 (3 x) - 4 \cdot - 4 + 6.25 x = 18.75$

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

Move $x$ .

$3 \cdot 3 (x \cdot x) + 3 x \cdot - 4 - 4 (3 x) - 4 \cdot - 4 + 6.25 x = 18.75$

Multiply $x$ by $x$ .

$3 \cdot 3 x^{2} + 3 x \cdot - 4 - 4 (3 x) - 4 \cdot - 4 + 6.25 x = 18.75$

Multiply $3$ by $3$ .

$9 x^{2} + 3 x \cdot - 4 - 4 (3 x) - 4 \cdot - 4 + 6.25 x = 18.75$

Multiply $- 4$ by $3$ .

$9 x^{2} - 12 x - 4 (3 x) - 4 \cdot - 4 + 6.25 x = 18.75$

Multiply $3$ by $- 4$ .

$9 x^{2} - 12 x - 12 x - 4 \cdot - 4 + 6.25 x = 18.75$

Multiply $- 4$ by $- 4$ .

$9 x^{2} - 12 x - 12 x + 16 + 6.25 x = 18.75$

Subtract $12 x$ from $- 12 x$ .

$9 x^{2} - 24 x + 16 + 6.25 x = 18.75$

Add $- 24 x$ and $6.25 x$ .

$9 x^{2} - 17.75 x + 16 = 18.75$

Subtract $18.75$ from both sides of the equation.

$9 x^{2} - 17.75 x + 16 - 18.75 = 0$

Subtract $18.75$ from $16$ .

$9 x^{2} - 17.75 x - 2.75 = 0$

Factor $0.25$ out of $9 x^{2} - 17.75 x - 2.75$ .

Factor $0.25$ out of $9 x^{2}$ .

$0.25 (36 x^{2}) - 17.75 x - 2.75 = 0$

Factor $0.25$ out of $- 17.75 x$ .

$0.25 (36 x^{2}) + 0.25 (- 71 x) - 2.75 = 0$

Factor $0.25$ out of $- 2.75$ .

$0.25 (36 x^{2}) + 0.25 (- 71 x) + 0.25 (- 11) = 0$

Factor $0.25$ out of $0.25 (36 x^{2}) + 0.25 (- 71 x)$ .

$0.25 (36 x^{2} - 71 x) + 0.25 (- 11) = 0$

Factor $0.25$ out of $0.25 (36 x^{2} - 71 x) + 0.25 (- 11)$ .

$0.25 (36 x^{2} - 71 x - 11) = 0$

Divide each term in $0.25 (36 x^{2} - 71 x - 11) = 0$ by $0.25$ and simplify.

Divide each term in $0.25 (36 x^{2} - 71 x - 11) = 0$ by $0.25$ .

$\frac{0.25 (36 x^{2} - 71 x - 11)}{0.25} = \frac{0}{0.25}$

Simplify the left side.

Cancel the common factor of $0.25$ .

Cancel the common factor.

$\frac{0.25 (36 x^{2} - 71 x - 11)}{0.25} = \frac{0}{0.25}$

Divide $36 x^{2} - 71 x - 11$ by $1$ .

$36 x^{2} - 71 x - 11 = \frac{0}{0.25}$

Simplify the right side.

Divide $0$ by $0.25$ .

$36 x^{2} - 71 x - 11 = 0$

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = 36$ , $b = - 71$ , and $c = - 11$ into the quadratic formula and solve for $x$ .

$\frac{71 \pm \sqrt{{(- 71)}^{2} - 4 \cdot (36 \cdot - 11)}}{2 \cdot 36}$

Simplify.

Simplify the numerator.

Raise $- 71$ to the power of $2$ .

$x = \frac{71 \pm \sqrt{5041 - 4 \cdot 36 \cdot - 11}}{2 \cdot 36}$

Multiply $- 4 \cdot 36 \cdot - 11$ .

Multiply $- 4$ by $36$ .

$x = \frac{71 \pm \sqrt{5041 - 144 \cdot - 11}}{2 \cdot 36}$

Multiply $- 144$ by $- 11$ .

$x = \frac{71 \pm \sqrt{5041 + 1584}}{2 \cdot 36}$

Add $5041$ and $1584$ .

$x = \frac{71 \pm \sqrt{6625}}{2 \cdot 36}$

Rewrite $6625$ as $5^{2} \cdot 265$ .

Factor $25$ out of $6625$ .

$x = \frac{71 \pm \sqrt{25 (265)}}{2 \cdot 36}$

Rewrite $25$ as $5^{2}$ .

$x = \frac{71 \pm \sqrt{5^{2} \cdot 265}}{2 \cdot 36}$

Pull terms out from under the radical.

$x = \frac{71 \pm 5 \sqrt{265}}{2 \cdot 36}$

Multiply $2$ by $36$ .

$x = \frac{71 \pm 5 \sqrt{265}}{72}$

The final answer is the combination of both solutions.

$x = \frac{71 + 5 \sqrt{265}}{72}, \frac{71 - 5 \sqrt{265}}{72}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \frac{71 + 5 \sqrt{265}}{72}, \frac{71 - 5 \sqrt{265}}{72}$

Decimal Form:

$x = 2.11658476 \dots, - 0.14436254 \dots$

Solve for x |3x-4|=2.5 square root of 3-x

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