Solve 49x^2+105x=98

$$49 { x }^{ 2 } +105x = 98$$

Solve for x

$$x=\frac{\sqrt{617}-15}{14}\approx 0.702820335$$

$$x=\frac{-\sqrt{617}-15}{14}\approx -2.845677478$$

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.

$$49x^{2}+105x=98$$

Subtract 98 from both sides of the equation.

$$49x^{2}+105x-98=98-98$$

Subtracting 98 from itself leaves 0.

$$49x^{2}+105x-98=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 49 for a, 105 for b, and -98 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-105±\sqrt{105^{2}-4\times 49\left(-98\right)}}{2\times 49}$$

Square 105.

$$x=\frac{-105±\sqrt{11025-4\times 49\left(-98\right)}}{2\times 49}$$

Multiply -4 times 49.

$$x=\frac{-105±\sqrt{11025-196\left(-98\right)}}{2\times 49}$$

Multiply -196 times -98.

$$x=\frac{-105±\sqrt{11025+19208}}{2\times 49}$$

Add 11025 to 19208.

$$x=\frac{-105±\sqrt{30233}}{2\times 49}$$

Take the square root of 30233.

$$x=\frac{-105±7\sqrt{617}}{2\times 49}$$

Multiply 2 times 49.

$$x=\frac{-105±7\sqrt{617}}{98}$$

Now solve the equation x=\frac{-105±7\sqrt{617}}{98} when ± is plus. Add -105 to 7\sqrt{617}.

$$x=\frac{7\sqrt{617}-105}{98}$$

Divide -105+7\sqrt{617} by 98.

$$x=\frac{\sqrt{617}-15}{14}$$

Now solve the equation x=\frac{-105±7\sqrt{617}}{98} when ± is minus. Subtract 7\sqrt{617} from -105.

$$x=\frac{-7\sqrt{617}-105}{98}$$

Divide -105-7\sqrt{617} by 98.

$$x=\frac{-\sqrt{617}-15}{14}$$

The equation is now solved.

$$x=\frac{\sqrt{617}-15}{14} x=\frac{-\sqrt{617}-15}{14}$$

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.

$$49x^{2}+105x=98$$

Divide both sides by 49.

$$\frac{49x^{2}+105x}{49}=\frac{98}{49}$$

Dividing by 49 undoes the multiplication by 49.

$$x^{2}+\frac{105}{49}x=\frac{98}{49}$$

Reduce the fraction \frac{105}{49} to lowest terms by extracting and canceling out 7.

$$x^{2}+\frac{15}{7}x=\frac{98}{49}$$

Divide 98 by 49.

$$x^{2}+\frac{15}{7}x=2$$

Divide \frac{15}{7}, the coefficient of the x term, by 2 to get \frac{15}{14}. Then add the square of \frac{15}{14} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{15}{7}x+\left(\frac{15}{14}\right)^{2}=2+\left(\frac{15}{14}\right)^{2}$$

Square \frac{15}{14} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{15}{7}x+\frac{225}{196}=2+\frac{225}{196}$$

Add 2 to \frac{225}{196}.

$$x^{2}+\frac{15}{7}x+\frac{225}{196}=\frac{617}{196}$$

Factor x^{2}+\frac{15}{7}x+\frac{225}{196}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{15}{14}\right)^{2}=\frac{617}{196}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{15}{14}\right)^{2}}=\sqrt{\frac{617}{196}}$$

Simplify.

$$x+\frac{15}{14}=\frac{\sqrt{617}}{14} x+\frac{15}{14}=-\frac{\sqrt{617}}{14}$$

Subtract \frac{15}{14} from both sides of the equation.

$$x=\frac{\sqrt{617}-15}{14} x=\frac{-\sqrt{617}-15}{14}$$