Solve Using the Quadratic Formula 14x^2+1=6x^2+7x

$14 x^{2} + 1 = 6 x^{2} + 7 x$

Step 1

Move all terms to the left side of the equation and simplify.

Move all the expressions to the left side of the equation.

Subtract $6 x^{2}$ from both sides of the equation.

$14 x^{2} + 1 - 6 x^{2} = 7 x$

Subtract $7 x$ from both sides of the equation.

$14 x^{2} + 1 - 6 x^{2} - 7 x = 0$

Subtract $6 x^{2}$ from $14 x^{2}$ .

$8 x^{2} + 1 - 7 x = 0$

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

Substitute the values $a = 8$ , $b = - 7$ , and $c = 1$ into the quadratic formula and solve for $x$ .

$\frac{7 \pm \sqrt{{(- 7)}^{2} - 4 \cdot (8 \cdot 1)}}{2 \cdot 8}$

Step 4

Simplify.

Simplify the numerator.

Raise $- 7$ to the power of $2$ .

$x = \frac{7 \pm \sqrt{49 - 4 \cdot 8 \cdot 1}}{2 \cdot 8}$

Multiply $- 4 \cdot 8 \cdot 1$ .

Multiply $- 4$ by $8$ .

$x = \frac{7 \pm \sqrt{49 - 32 \cdot 1}}{2 \cdot 8}$

Multiply $- 32$ by $1$ .

$x = \frac{7 \pm \sqrt{49 - 32}}{2 \cdot 8}$

Subtract $32$ from $49$ .

$x = \frac{7 \pm \sqrt{17}}{2 \cdot 8}$

Multiply $2$ by $8$ .

$x = \frac{7 \pm \sqrt{17}}{16}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{7 \pm \sqrt{17}}{16}$

Decimal Form:

$x = 0.69519410 \dots, 0.17980589 \dots$