Solve 2x*(x+7)=3

$$2 x \cdot ( x + 7 ) = 3$$

Solve for x

$$x=\frac{\sqrt{55}-7}{2}\approx 0.208099244$$

$$x=\frac{-\sqrt{55}-7}{2}\approx -7.208099244$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 2x by x+7.

$$2x^{2}+14x=3$$

Subtract 3 from both sides.

$$2x^{2}+14x-3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 14 for b, and -3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-14±\sqrt{14^{2}-4\times 2\left(-3\right)}}{2\times 2}$$

Square 14.

$$x=\frac{-14±\sqrt{196-4\times 2\left(-3\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-14±\sqrt{196-8\left(-3\right)}}{2\times 2}$$

Multiply -8 times -3.

$$x=\frac{-14±\sqrt{196+24}}{2\times 2}$$

Add 196 to 24.

$$x=\frac{-14±\sqrt{220}}{2\times 2}$$

Take the square root of 220.

$$x=\frac{-14±2\sqrt{55}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-14±2\sqrt{55}}{4}$$

Now solve the equation x=\frac{-14±2\sqrt{55}}{4} when ± is plus. Add -14 to 2\sqrt{55}.

$$x=\frac{2\sqrt{55}-14}{4}$$

Divide -14+2\sqrt{55} by 4.

$$x=\frac{\sqrt{55}-7}{2}$$

Now solve the equation x=\frac{-14±2\sqrt{55}}{4} when ± is minus. Subtract 2\sqrt{55} from -14.

$$x=\frac{-2\sqrt{55}-14}{4}$$

Divide -14-2\sqrt{55} by 4.

$$x=\frac{-\sqrt{55}-7}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{55}-7}{2} x=\frac{-\sqrt{55}-7}{2}$$

Steps for Completing the Square

Use the distributive property to multiply 2x by x+7.

$$2x^{2}+14x=3$$

Divide both sides by 2.

$$\frac{2x^{2}+14x}{2}=\frac{3}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{14}{2}x=\frac{3}{2}$$

Divide 14 by 2.

$$x^{2}+7x=\frac{3}{2}$$

Divide 7, the coefficient of the x term, by 2 to get \frac{7}{2}. Then add the square of \frac{7}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+7x+\left(\frac{7}{2}\right)^{2}=\frac{3}{2}+\left(\frac{7}{2}\right)^{2}$$

Square \frac{7}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+7x+\frac{49}{4}=\frac{3}{2}+\frac{49}{4}$$

Add \frac{3}{2} to \frac{49}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+7x+\frac{49}{4}=\frac{55}{4}$$

Factor x^{2}+7x+\frac{49}{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}.

$$\left(x+\frac{7}{2}\right)^{2}=\frac{55}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{7}{2}\right)^{2}}=\sqrt{\frac{55}{4}}$$

Simplify.

$$x+\frac{7}{2}=\frac{\sqrt{55}}{2} x+\frac{7}{2}=-\frac{\sqrt{55}}{2}$$

Subtract \frac{7}{2} from both sides of the equation.

$$x=\frac{\sqrt{55}-7}{2} x=\frac{-\sqrt{55}-7}{2}$$