Solve 3x^2+1=2x+7

$$3 x ^ { 2 } + 1 = 2 x + 7$$

Solve for x

$$x = \frac{\sqrt{19} + 1}{3} \approx 1.786299648$$

$$x=\frac{1-\sqrt{19}}{3}\approx -1.119632981$$

Steps Using the Quadratic Formula

Subtract 2x from both sides.

$$3x^{2}+1-2x=7$$

Subtract 7 from both sides.

$$3x^{2}+1-2x-7=0$$

Subtract 7 from 1 to get -6.

$$3x^{2}-6-2x=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$3x^{2}-2x-6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -2 for b, and -6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\times 3\left(-6\right)}}{2\times 3}$$

Square -2.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\times 3\left(-6\right)}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-\left(-2\right)±\sqrt{4-12\left(-6\right)}}{2\times 3}$$

Multiply -12 times -6.

$$x=\frac{-\left(-2\right)±\sqrt{4+72}}{2\times 3}$$

Add 4 to 72.

$$x=\frac{-\left(-2\right)±\sqrt{76}}{2\times 3}$$

Take the square root of 76.

$$x=\frac{-\left(-2\right)±2\sqrt{19}}{2\times 3}$$

The opposite of -2 is 2.

$$x=\frac{2±2\sqrt{19}}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{2±2\sqrt{19}}{6}$$

Now solve the equation x=\frac{2±2\sqrt{19}}{6} when ± is plus. Add 2 to 2\sqrt{19}.

$$x=\frac{2\sqrt{19}+2}{6}$$

Divide 2+2\sqrt{19} by 6.

$$x=\frac{\sqrt{19}+1}{3}$$

Now solve the equation x=\frac{2±2\sqrt{19}}{6} when ± is minus. Subtract 2\sqrt{19} from 2.

$$x=\frac{2-2\sqrt{19}}{6}$$

Divide 2-2\sqrt{19} by 6.

$$x=\frac{1-\sqrt{19}}{3}$$

The equation is now solved.

$$x=\frac{\sqrt{19}+1}{3} x=\frac{1-\sqrt{19}}{3}$$

Steps for Completing the Square

Subtract 2x from both sides.

$$3x^{2}+1-2x=7$$

Subtract 1 from both sides.

$$3x^{2}-2x=7-1$$

Subtract 1 from 7 to get 6.

$$3x^{2}-2x=6$$

Divide both sides by 3.

$$\frac{3x^{2}-2x}{3}=\frac{6}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$x^{2}-\frac{2}{3}x=\frac{6}{3}$$

Divide 6 by 3.

$$x^{2}-\frac{2}{3}x=2$$

Divide -\frac{2}{3}, the coefficient of the x term, by 2 to get -\frac{1}{3}. Then add the square of -\frac{1}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{2}{3}x+\left(-\frac{1}{3}\right)^{2}=2+\left(-\frac{1}{3}\right)^{2}$$

Square -\frac{1}{3} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{2}{3}x+\frac{1}{9}=2+\frac{1}{9}$$

Add 2 to \frac{1}{9}.

$$x^{2}-\frac{2}{3}x+\frac{1}{9}=\frac{19}{9}$$

Factor x^{2}-\frac{2}{3}x+\frac{1}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{3}\right)^{2}=\frac{19}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{3}\right)^{2}}=\sqrt{\frac{19}{9}}$$

Simplify.

$$x-\frac{1}{3}=\frac{\sqrt{19}}{3} x-\frac{1}{3}=-\frac{\sqrt{19}}{3}$$

Add \frac{1}{3} to both sides of the equation.

$$x=\frac{\sqrt{19}+1}{3} x=\frac{1-\sqrt{19}}{3}$$