Solve (4.2+2x)(3.5+2x)-14.7=18.8

$$(4.2+2x)(3.5+2x)-14.7=18.8$$

Solve for x

$$x=\frac{\sqrt{13449}-77}{40}\approx 0.974245592$$

$$x=\frac{-\sqrt{13449}-77}{40}\approx -4.824245592$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 4.2+2x by 3.5+2x and combine like terms.

$$14.7+15.4x+4x^{2}-14.7=18.8$$

Subtract 14.7 from 14.7 to get 0.

$$15.4x+4x^{2}=18.8$$

Subtract 18.8 from both sides.

$$15.4x+4x^{2}-18.8=0$$

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.

$$4x^{2}+15.4x-18.8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 15.4 for b, and -18.8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-15.4±\sqrt{15.4^{2}-4\times 4\left(-18.8\right)}}{2\times 4}$$

Square 15.4 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-15.4±\sqrt{237.16-4\times 4\left(-18.8\right)}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-15.4±\sqrt{237.16-16\left(-18.8\right)}}{2\times 4}$$

Multiply -16 times -18.8.

$$x=\frac{-15.4±\sqrt{237.16+300.8}}{2\times 4}$$

Add 237.16 to 300.8 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-15.4±\sqrt{537.96}}{2\times 4}$$

Take the square root of 537.96.

$$x=\frac{-15.4±\frac{\sqrt{13449}}{5}}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{-15.4±\frac{\sqrt{13449}}{5}}{8}$$

Now solve the equation x=\frac{-15.4±\frac{\sqrt{13449}}{5}}{8} when ± is plus. Add -15.4 to \frac{\sqrt{13449}}{5}.

$$x=\frac{\sqrt{13449}-77}{5\times 8}$$

Divide \frac{-77+\sqrt{13449}}{5} by 8.

$$x=\frac{\sqrt{13449}-77}{40}$$

Now solve the equation x=\frac{-15.4±\frac{\sqrt{13449}}{5}}{8} when ± is minus. Subtract \frac{\sqrt{13449}}{5} from -15.4.

$$x=\frac{-\sqrt{13449}-77}{5\times 8}$$

Divide \frac{-77-\sqrt{13449}}{5} by 8.

$$x=\frac{-\sqrt{13449}-77}{40}$$

The equation is now solved.

$$x=\frac{\sqrt{13449}-77}{40} x=\frac{-\sqrt{13449}-77}{40}$$

Steps for Completing the Square

Use the distributive property to multiply 4.2+2x by 3.5+2x and combine like terms.

$$14.7+15.4x+4x^{2}-14.7=18.8$$

Subtract 14.7 from 14.7 to get 0.

$$15.4x+4x^{2}=18.8$$

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.

$$4x^{2}+15.4x=18.8$$

Divide both sides by 4.

$$\frac{4x^{2}+15.4x}{4}=\frac{18.8}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\frac{15.4}{4}x=\frac{18.8}{4}$$

Divide 15.4 by 4.

$$x^{2}+3.85x=\frac{18.8}{4}$$

Divide 18.8 by 4.

$$x^{2}+3.85x=4.7$$

Divide 3.85, the coefficient of the x term, by 2 to get 1.925. Then add the square of 1.925 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+3.85x+1.925^{2}=4.7+1.925^{2}$$

Square 1.925 by squaring both the numerator and the denominator of the fraction.

$$x^{2}+3.85x+3.705625=4.7+3.705625$$

Add 4.7 to 3.705625 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+3.85x+3.705625=8.405625$$

Factor x^{2}+3.85x+3.705625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+1.925\right)^{2}=8.405625$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1.925\right)^{2}}=\sqrt{8.405625}$$

Simplify.

$$x+1.925=\frac{\sqrt{13449}}{40} x+1.925=-\frac{\sqrt{13449}}{40}$$

Subtract 1.925 from both sides of the equation.

$$x=\frac{\sqrt{13449}-77}{40} x=\frac{-\sqrt{13449}-77}{40}$$