Solve 14x^2-84+60x=0

$$14 x ^ { 2 } - 84 + 60 x = 0$$

Solve for x

$$x = \frac{\sqrt{519} - 15}{7} \approx 1.111653071$$

$$x=\frac{-\sqrt{519}-15}{7}\approx -5.397367357$$

Steps Using the Quadratic Formula

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.

$$14x^{2}+60x-84=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 14 for a, 60 for b, and -84 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-60±\sqrt{60^{2}-4\times 14\left(-84\right)}}{2\times 14}$$

Square 60.

$$x=\frac{-60±\sqrt{3600-4\times 14\left(-84\right)}}{2\times 14}$$

Multiply -4 times 14.

$$x=\frac{-60±\sqrt{3600-56\left(-84\right)}}{2\times 14}$$

Multiply -56 times -84.

$$x=\frac{-60±\sqrt{3600+4704}}{2\times 14}$$

Add 3600 to 4704.

$$x=\frac{-60±\sqrt{8304}}{2\times 14}$$

Take the square root of 8304.

$$x=\frac{-60±4\sqrt{519}}{2\times 14}$$

Multiply 2 times 14.

$$x=\frac{-60±4\sqrt{519}}{28}$$

Now solve the equation x=\frac{-60±4\sqrt{519}}{28} when ± is plus. Add -60 to 4\sqrt{519}.

$$x=\frac{4\sqrt{519}-60}{28}$$

Divide -60+4\sqrt{519} by 28.

$$x=\frac{\sqrt{519}-15}{7}$$

Now solve the equation x=\frac{-60±4\sqrt{519}}{28} when ± is minus. Subtract 4\sqrt{519} from -60.

$$x=\frac{-4\sqrt{519}-60}{28}$$

Divide -60-4\sqrt{519} by 28.

$$x=\frac{-\sqrt{519}-15}{7}$$

The equation is now solved.

$$x=\frac{\sqrt{519}-15}{7} x=\frac{-\sqrt{519}-15}{7}$$

Steps for Completing the Square

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.

$$14x^{2}+60x-84=0$$

Add 84 to both sides of the equation.

$$14x^{2}+60x-84-\left(-84\right)=-\left(-84\right)$$

Subtracting -84 from itself leaves 0.

$$14x^{2}+60x=-\left(-84\right)$$

Subtract -84 from 0.

$$14x^{2}+60x=84$$

Divide both sides by 14.

$$\frac{14x^{2}+60x}{14}=\frac{84}{14}$$

Dividing by 14 undoes the multiplication by 14.

$$x^{2}+\frac{60}{14}x=\frac{84}{14}$$

Reduce the fraction \frac{60}{14} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{30}{7}x=\frac{84}{14}$$

Divide 84 by 14.

$$x^{2}+\frac{30}{7}x=6$$

Divide \frac{30}{7}, the coefficient of the x term, by 2 to get \frac{15}{7}. Then add the square of \frac{15}{7} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{30}{7}x+\left(\frac{15}{7}\right)^{2}=6+\left(\frac{15}{7}\right)^{2}$$

Square \frac{15}{7} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{30}{7}x+\frac{225}{49}=6+\frac{225}{49}$$

Add 6 to \frac{225}{49}.

$$x^{2}+\frac{30}{7}x+\frac{225}{49}=\frac{519}{49}$$

Factor x^{2}+\frac{30}{7}x+\frac{225}{49}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{15}{7}\right)^{2}=\frac{519}{49}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{15}{7}\right)^{2}}=\sqrt{\frac{519}{49}}$$

Simplify.

$$x+\frac{15}{7}=\frac{\sqrt{519}}{7} x+\frac{15}{7}=-\frac{\sqrt{519}}{7}$$

Subtract \frac{15}{7} from both sides of the equation.

$$x=\frac{\sqrt{519}-15}{7} x=\frac{-\sqrt{519}-15}{7}$$