Solve 4x+3x+7x=18x^2

$$4x+3x+7x=18 { x }^{ 2 }$$

Solve for x

$$x=\frac{7}{9}\approx 0.777777778$$

$$x=0$$

Steps Using Factoring

Combine 4x and 3x to get 7x.

$$7x+7x=18x^{2}$$

Combine 7x and 7x to get 14x.

$$14x=18x^{2}$$

Subtract 18x^{2} from both sides.

$$14x-18x^{2}=0$$

Factor out x.

$$x\left(14-18x\right)=0$$

To find equation solutions, solve x=0 and 14-18x=0.

$$x=0 x=\frac{7}{9}$$

Steps Using the Quadratic Formula

Combine 4x and 3x to get 7x.

$$7x+7x=18x^{2}$$

Combine 7x and 7x to get 14x.

$$14x=18x^{2}$$

Subtract 18x^{2} from both sides.

$$14x-18x^{2}=0$$

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.

$$-18x^{2}+14x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -18 for a, 14 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-14±\sqrt{14^{2}}}{2\left(-18\right)}$$

Take the square root of 14^{2}.

$$x=\frac{-14±14}{2\left(-18\right)}$$

Multiply 2 times -18.

$$x=\frac{-14±14}{-36}$$

Now solve the equation x=\frac{-14±14}{-36} when ± is plus. Add -14 to 14.

$$x=\frac{0}{-36}$$

Divide 0 by -36.

$$x=0$$

Now solve the equation x=\frac{-14±14}{-36} when ± is minus. Subtract 14 from -14.

$$x=-\frac{28}{-36}$$

Reduce the fraction \frac{-28}{-36} to lowest terms by extracting and canceling out 4.

$$x=\frac{7}{9}$$

The equation is now solved.

$$x=0 x=\frac{7}{9}$$

Steps for Completing the Square

Combine 4x and 3x to get 7x.

$$7x+7x=18x^{2}$$

Combine 7x and 7x to get 14x.

$$14x=18x^{2}$$

Subtract 18x^{2} from both sides.

$$14x-18x^{2}=0$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-18x^{2}+14x=0$$

Divide both sides by -18.

$$\frac{-18x^{2}+14x}{-18}=\frac{0}{-18}$$

Dividing by -18 undoes the multiplication by -18.

$$x^{2}+\frac{14}{-18}x=\frac{0}{-18}$$

Reduce the fraction \frac{14}{-18} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{7}{9}x=\frac{0}{-18}$$

Divide 0 by -18.

$$x^{2}-\frac{7}{9}x=0$$

Divide -\frac{7}{9}, the coefficient of the x term, by 2 to get -\frac{7}{18}. Then add the square of -\frac{7}{18} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{7}{9}x+\left(-\frac{7}{18}\right)^{2}=\left(-\frac{7}{18}\right)^{2}$$

Square -\frac{7}{18} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{7}{9}x+\frac{49}{324}=\frac{49}{324}$$

Factor x^{2}-\frac{7}{9}x+\frac{49}{324}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{7}{18}\right)^{2}=\frac{49}{324}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7}{18}\right)^{2}}=\sqrt{\frac{49}{324}}$$

Simplify.

$$x-\frac{7}{18}=\frac{7}{18} x-\frac{7}{18}=-\frac{7}{18}$$

Add \frac{7}{18} to both sides of the equation.

$$x=\frac{7}{9} x=0$$