Solve x^2=19+3x

$$x ^ { 2 } = 19 + 3 x$$

Solve for x

$$x = \frac{\sqrt{85} + 3}{2} \approx 6.109772229$$

$$x=\frac{3-\sqrt{85}}{2}\approx -3.109772229$$

Steps Using the Quadratic Formula

Subtract 19 from both sides.

$$x^{2}-19=3x$$

Subtract 3x from both sides.

$$x^{2}-19-3x=0$$

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.

$$x^{2}-3x-19=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -3 for b, and -19 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\left(-19\right)}}{2}$$

Square -3.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\left(-19\right)}}{2}$$

Multiply -4 times -19.

$$x=\frac{-\left(-3\right)±\sqrt{9+76}}{2}$$

Add 9 to 76.

$$x=\frac{-\left(-3\right)±\sqrt{85}}{2}$$

The opposite of -3 is 3.

$$x=\frac{3±\sqrt{85}}{2}$$

Now solve the equation x=\frac{3±\sqrt{85}}{2} when ± is plus. Add 3 to \sqrt{85}.

$$x=\frac{\sqrt{85}+3}{2}$$

Now solve the equation x=\frac{3±\sqrt{85}}{2} when ± is minus. Subtract \sqrt{85} from 3.

$$x=\frac{3-\sqrt{85}}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{85}+3}{2} x=\frac{3-\sqrt{85}}{2}$$

Steps for Completing the Square

Subtract 3x from both sides.

$$x^{2}-3x=19$$

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.

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

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

$$x^{2}-3x+\frac{9}{4}=19+\frac{9}{4}$$

Add 19 to \frac{9}{4}.

$$x^{2}-3x+\frac{9}{4}=\frac{85}{4}$$

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}.

$$\left(x-\frac{3}{2}\right)^{2}=\frac{85}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{2}\right)^{2}}=\sqrt{\frac{85}{4}}$$

Simplify.

$$x-\frac{3}{2}=\frac{\sqrt{85}}{2} x-\frac{3}{2}=-\frac{\sqrt{85}}{2}$$

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

$$x=\frac{\sqrt{85}+3}{2} x=\frac{3-\sqrt{85}}{2}$$