Solve x^2=3x+47

$$x ^ { 2 } = 3 x + 47$$

Solve for x

$$x = \frac{\sqrt{197} + 3}{2} \approx 8.517834424$$

$$x=\frac{3-\sqrt{197}}{2}\approx -5.517834424$$

Steps Using the Quadratic Formula

Subtract 3x from both sides.

$$x^{2}-3x=47$$

Subtract 47 from both sides.

$$x^{2}-3x-47=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -3 for b, and -47 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\left(-47\right)}}{2}$$

Square -3.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\left(-47\right)}}{2}$$

Multiply -4 times -47.

$$x=\frac{-\left(-3\right)±\sqrt{9+188}}{2}$$

Add 9 to 188.

$$x=\frac{-\left(-3\right)±\sqrt{197}}{2}$$

The opposite of -3 is 3.

$$x=\frac{3±\sqrt{197}}{2}$$

Now solve the equation x=\frac{3±\sqrt{197}}{2} when ± is plus. Add 3 to \sqrt{197}.

$$x=\frac{\sqrt{197}+3}{2}$$

Now solve the equation x=\frac{3±\sqrt{197}}{2} when ± is minus. Subtract \sqrt{197} from 3.

$$x=\frac{3-\sqrt{197}}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{197}+3}{2} x=\frac{3-\sqrt{197}}{2}$$

Steps for Completing the Square

Subtract 3x from both sides.

$$x^{2}-3x=47$$

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

$$x^{2}-3x+\left(-\frac{3}{2}\right)^{2}=47+\left(-\frac{3}{2}\right)^{2}$$

Square -\frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-3x+\frac{9}{4}=47+\frac{9}{4}$$

Add 47 to \frac{9}{4}.

$$x^{2}-3x+\frac{9}{4}=\frac{197}{4}$$

Factor x^{2}-3x+\frac{9}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{3}{2}\right)^{2}=\frac{197}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{2}\right)^{2}}=\sqrt{\frac{197}{4}}$$

Simplify.

$$x-\frac{3}{2}=\frac{\sqrt{197}}{2} x-\frac{3}{2}=-\frac{\sqrt{197}}{2}$$

Add \frac{3}{2} to both sides of the equation.

$$x=\frac{\sqrt{197}+3}{2} x=\frac{3-\sqrt{197}}{2}$$