Solve for x 3x^2+174=48x

$3 x^{2} + 174 = 48 x$

Step 1

Subtract $48 x$ from both sides of the equation.

$3 x^{2} + 174 - 48 x = 0$

Step 2

Factor the left side of the equation.

Factor $3$ out of $3 x^{2} + 174 - 48 x$ .

Factor $3$ out of $3 x^{2}$ .

$3 (x^{2}) + 174 - 48 x = 0$

Factor $3$ out of $174$ .

$3 x^{2} + 3 \cdot 58 - 48 x = 0$

Factor $3$ out of $- 48 x$ .

$3 x^{2} + 3 \cdot 58 + 3 (- 16 x) = 0$

Factor $3$ out of $3 x^{2} + 3 \cdot 58$ .

$3 (x^{2} + 58) + 3 (- 16 x) = 0$

Factor $3$ out of $3 (x^{2} + 58) + 3 (- 16 x)$ .

$3 (x^{2} + 58 - 16 x) = 0$

Reorder terms.

$3 (x^{2} - 16 x + 58) = 0$

Step 3

Divide each term in $3 (x^{2} - 16 x + 58) = 0$ by $3$ and simplify.

Divide each term in $3 (x^{2} - 16 x + 58) = 0$ by $3$ .

$\frac{3 (x^{2} - 16 x + 58)}{3} = \frac{0}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 (x^{2} - 16 x + 58)}{3} = \frac{0}{3}$

Divide $x^{2} - 16 x + 58$ by $1$ .

$x^{2} - 16 x + 58 = \frac{0}{3}$

Simplify the right side.

Divide $0$ by $3$ .

$x^{2} - 16 x + 58 = 0$

Step 4

Use the quadratic formula to find the solutions.

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

Step 5

Substitute the values $a = 1$ , $b = - 16$ , and $c = 58$ into the quadratic formula and solve for $x$ .

$\frac{16 \pm \sqrt{{(- 16)}^{2} - 4 \cdot (1 \cdot 58)}}{2 \cdot 1}$

Step 6

Simplify.

Simplify the numerator.

Raise $- 16$ to the power of $2$ .

$x = \frac{16 \pm \sqrt{256 - 4 \cdot 1 \cdot 58}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 58$ .

Multiply $- 4$ by $1$ .

$x = \frac{16 \pm \sqrt{256 - 4 \cdot 58}}{2 \cdot 1}$

Multiply $- 4$ by $58$ .

$x = \frac{16 \pm \sqrt{256 - 232}}{2 \cdot 1}$

Subtract $232$ from $256$ .

$x = \frac{16 \pm \sqrt{24}}{2 \cdot 1}$

Rewrite $24$ as $2^{2} \cdot 6$ .

Factor $4$ out of $24$ .

$x = \frac{16 \pm \sqrt{4 (6)}}{2 \cdot 1}$

Rewrite $4$ as $2^{2}$ .

$x = \frac{16 \pm \sqrt{2^{2} \cdot 6}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{16 \pm 2 \sqrt{6}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{16 \pm 2 \sqrt{6}}{2}$

Simplify $\frac{16 \pm 2 \sqrt{6}}{2}$ .

$x = 8 \pm \sqrt{6}$

Step 7

The final answer is the combination of both solutions.

$x = 8 + \sqrt{6}, 8 - \sqrt{6}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = 8 + \sqrt{6}, 8 - \sqrt{6}$

Decimal Form:

$x = 10.44948974 \dots, 5.55051025 \dots$

Solve for x 3x^2+174=48x

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