Solve 65x^2+126x+49=0

$$65 { x }^{ 2 } +126x+49 = 0$$

Solve for x

$$x=-\frac{7}{13}\approx -0.538461538$$

$$x = -\frac{7}{5} = -1\frac{2}{5} = -1.4$$

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.

$$65x^{2}+126x+49=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 65 for a, 126 for b, and 49 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-126±\sqrt{126^{2}-4\times 65\times 49}}{2\times 65}$$

Square 126.

$$x=\frac{-126±\sqrt{15876-4\times 65\times 49}}{2\times 65}$$

Multiply -4 times 65.

$$x=\frac{-126±\sqrt{15876-260\times 49}}{2\times 65}$$

Multiply -260 times 49.

$$x=\frac{-126±\sqrt{15876-12740}}{2\times 65}$$

Add 15876 to -12740.

$$x=\frac{-126±\sqrt{3136}}{2\times 65}$$

Take the square root of 3136.

$$x=\frac{-126±56}{2\times 65}$$

Multiply 2 times 65.

$$x=\frac{-126±56}{130}$$

Now solve the equation x=\frac{-126±56}{130} when ± is plus. Add -126 to 56.

$$x=-\frac{70}{130}$$

Reduce the fraction \frac{-70}{130} to lowest terms by extracting and canceling out 10.

$$x=-\frac{7}{13}$$

Now solve the equation x=\frac{-126±56}{130} when ± is minus. Subtract 56 from -126.

$$x=-\frac{182}{130}$$

Reduce the fraction \frac{-182}{130} to lowest terms by extracting and canceling out 26.

$$x=-\frac{7}{5}$$

The equation is now solved.

$$x=-\frac{7}{13} x=-\frac{7}{5}$$

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.

$$65x^{2}+126x+49=0$$

Subtract 49 from both sides of the equation.

$$65x^{2}+126x+49-49=-49$$

Subtracting 49 from itself leaves 0.

$$65x^{2}+126x=-49$$

Divide both sides by 65.

$$\frac{65x^{2}+126x}{65}=-\frac{49}{65}$$

Dividing by 65 undoes the multiplication by 65.

$$x^{2}+\frac{126}{65}x=-\frac{49}{65}$$

Divide \frac{126}{65}, the coefficient of the x term, by 2 to get \frac{63}{65}. Then add the square of \frac{63}{65} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{126}{65}x+\left(\frac{63}{65}\right)^{2}=-\frac{49}{65}+\left(\frac{63}{65}\right)^{2}$$

Square \frac{63}{65} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{126}{65}x+\frac{3969}{4225}=-\frac{49}{65}+\frac{3969}{4225}$$

Add -\frac{49}{65} to \frac{3969}{4225} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{126}{65}x+\frac{3969}{4225}=\frac{784}{4225}$$

Factor x^{2}+\frac{126}{65}x+\frac{3969}{4225}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{63}{65}\right)^{2}=\frac{784}{4225}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{63}{65}\right)^{2}}=\sqrt{\frac{784}{4225}}$$

Simplify.

$$x+\frac{63}{65}=\frac{28}{65} x+\frac{63}{65}=-\frac{28}{65}$$

Subtract \frac{63}{65} from both sides of the equation.

$$x=-\frac{7}{13} x=-\frac{7}{5}$$