Solve 3x^2+10x-1=2x^2+4x+15.3

$$3 x ^ { 2 } + 10 x - 1 = 2 x ^ { 2 } + 4 x + 15.3$$

Solve for x

$$x=\frac{\sqrt{2530}}{10}-3\approx 2.029910536$$

$$x=-\frac{\sqrt{2530}}{10}-3\approx -8.029910536$$

Steps Using the Quadratic Formula

Subtract 2x^{2} from both sides.

$$3x^{2}+10x-1-2x^{2}=4x+15.3$$

Combine 3x^{2} and -2x^{2} to get x^{2}.

$$x^{2}+10x-1=4x+15.3$$

Subtract 4x from both sides.

$$x^{2}+10x-1-4x=15.3$$

Combine 10x and -4x to get 6x.

$$x^{2}+6x-1=15.3$$

Subtract 15.3 from both sides.

$$x^{2}+6x-1-15.3=0$$

Subtract 15.3 from -1 to get -16.3.

$$x^{2}+6x-16.3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 6 for b, and -16.3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-6±\sqrt{6^{2}-4\left(-16.3\right)}}{2}$$

Square 6.

$$x=\frac{-6±\sqrt{36-4\left(-16.3\right)}}{2}$$

Multiply -4 times -16.3.

$$x=\frac{-6±\sqrt{36+65.2}}{2}$$

Add 36 to 65.2.

$$x=\frac{-6±\sqrt{101.2}}{2}$$

Take the square root of 101.2.

$$x=\frac{-6±\frac{\sqrt{2530}}{5}}{2}$$

Now solve the equation x=\frac{-6±\frac{\sqrt{2530}}{5}}{2} when ± is plus. Add -6 to \frac{\sqrt{2530}}{5}.

$$x=\frac{\frac{\sqrt{2530}}{5}-6}{2}$$

Divide -6+\frac{\sqrt{2530}}{5} by 2.

$$x=\frac{\sqrt{2530}}{10}-3$$

Now solve the equation x=\frac{-6±\frac{\sqrt{2530}}{5}}{2} when ± is minus. Subtract \frac{\sqrt{2530}}{5} from -6.

$$x=\frac{-\frac{\sqrt{2530}}{5}-6}{2}$$

Divide -6-\frac{\sqrt{2530}}{5} by 2.

$$x=-\frac{\sqrt{2530}}{10}-3$$

The equation is now solved.

$$x=\frac{\sqrt{2530}}{10}-3 x=-\frac{\sqrt{2530}}{10}-3$$

Steps for Completing the Square

Subtract 2x^{2} from both sides.

$$3x^{2}+10x-1-2x^{2}=4x+15.3$$

Combine 3x^{2} and -2x^{2} to get x^{2}.

$$x^{2}+10x-1=4x+15.3$$

Subtract 4x from both sides.

$$x^{2}+10x-1-4x=15.3$$

Combine 10x and -4x to get 6x.

$$x^{2}+6x-1=15.3$$

Add 1 to both sides.

$$x^{2}+6x=15.3+1$$

Add 15.3 and 1 to get 16.3.

$$x^{2}+6x=16.3$$

Divide 6, the coefficient of the x term, by 2 to get 3. Then add the square of 3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+6x+3^{2}=16.3+3^{2}$$

Square 3.

$$x^{2}+6x+9=16.3+9$$

Add 16.3 to 9.

$$x^{2}+6x+9=25.3$$

Factor x^{2}+6x+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}.

$$\left(x+3\right)^{2}=25.3$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+3\right)^{2}}=\sqrt{25.3}$$

Simplify.

$$x+3=\frac{\sqrt{2530}}{10} x+3=-\frac{\sqrt{2530}}{10}$$

Subtract 3 from both sides of the equation.

$$x=\frac{\sqrt{2530}}{10}-3 x=-\frac{\sqrt{2530}}{10}-3$$