Solve 1/2x^2-2/7=5/14x

$$\frac { 1 } { 2 } x ^ { 2 } - \frac { 2 } { 7 } = \frac { 5 } { 14 } x$$

Solve for x

$$x = \frac{\sqrt{137} + 5}{14} \approx 1.193192851$$

$$x=\frac{5-\sqrt{137}}{14}\approx -0.478907136$$

Steps Using the Quadratic Formula

Subtract \frac{5}{14}x from both sides.

$$\frac{1}{2}x^{2}-\frac{2}{7}-\frac{5}{14}x=0$$

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.

$$\frac{1}{2}x^{2}-\frac{5}{14}x-\frac{2}{7}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{1}{2} for a, -\frac{5}{14} for b, and -\frac{2}{7} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{5}{14}\right)±\sqrt{\left(-\frac{5}{14}\right)^{2}-4\times \frac{1}{2}\left(-\frac{2}{7}\right)}}{2\times \frac{1}{2}}$$

Square -\frac{5}{14} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-\frac{5}{14}\right)±\sqrt{\frac{25}{196}-4\times \frac{1}{2}\left(-\frac{2}{7}\right)}}{2\times \frac{1}{2}}$$

Multiply -4 times \frac{1}{2}.

$$x=\frac{-\left(-\frac{5}{14}\right)±\sqrt{\frac{25}{196}-2\left(-\frac{2}{7}\right)}}{2\times \frac{1}{2}}$$

Multiply -2 times -\frac{2}{7}.

$$x=\frac{-\left(-\frac{5}{14}\right)±\sqrt{\frac{25}{196}+\frac{4}{7}}}{2\times \frac{1}{2}}$$

Add \frac{25}{196} to \frac{4}{7} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-\frac{5}{14}\right)±\sqrt{\frac{137}{196}}}{2\times \frac{1}{2}}$$

Take the square root of \frac{137}{196}.

$$x=\frac{-\left(-\frac{5}{14}\right)±\frac{\sqrt{137}}{14}}{2\times \frac{1}{2}}$$

The opposite of -\frac{5}{14} is \frac{5}{14}.

$$x=\frac{\frac{5}{14}±\frac{\sqrt{137}}{14}}{2\times \frac{1}{2}}$$

Multiply 2 times \frac{1}{2}.

$$x=\frac{\frac{5}{14}±\frac{\sqrt{137}}{14}}{1}$$

Now solve the equation x=\frac{\frac{5}{14}±\frac{\sqrt{137}}{14}}{1} when ± is plus. Add \frac{5}{14} to \frac{\sqrt{137}}{14}.

$$x=\frac{\sqrt{137}+5}{14}$$

Now solve the equation x=\frac{\frac{5}{14}±\frac{\sqrt{137}}{14}}{1} when ± is minus. Subtract \frac{\sqrt{137}}{14} from \frac{5}{14}.

$$x=\frac{5-\sqrt{137}}{14}$$

The equation is now solved.

$$x=\frac{\sqrt{137}+5}{14} x=\frac{5-\sqrt{137}}{14}$$

Steps for Completing the Square

Subtract \frac{5}{14}x from both sides.

$$\frac{1}{2}x^{2}-\frac{2}{7}-\frac{5}{14}x=0$$

Add \frac{2}{7} to both sides. Anything plus zero gives itself.

$$\frac{1}{2}x^{2}-\frac{5}{14}x=\frac{2}{7}$$

Multiply both sides by 2.

$$\frac{\frac{1}{2}x^{2}-\frac{5}{14}x}{\frac{1}{2}}=\frac{\frac{2}{7}}{\frac{1}{2}}$$

Dividing by \frac{1}{2} undoes the multiplication by \frac{1}{2}.

$$x^{2}+\left(-\frac{\frac{5}{14}}{\frac{1}{2}}\right)x=\frac{\frac{2}{7}}{\frac{1}{2}}$$

Divide -\frac{5}{14} by \frac{1}{2} by multiplying -\frac{5}{14} by the reciprocal of \frac{1}{2}.

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

Divide \frac{2}{7} by \frac{1}{2} by multiplying \frac{2}{7} by the reciprocal of \frac{1}{2}.

$$x^{2}-\frac{5}{7}x=\frac{4}{7}$$

Divide -\frac{5}{7}, the coefficient of the x term, by 2 to get -\frac{5}{14}. Then add the square of -\frac{5}{14} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{5}{7}x+\left(-\frac{5}{14}\right)^{2}=\frac{4}{7}+\left(-\frac{5}{14}\right)^{2}$$

Square -\frac{5}{14} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{5}{7}x+\frac{25}{196}=\frac{4}{7}+\frac{25}{196}$$

Add \frac{4}{7} to \frac{25}{196} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{5}{7}x+\frac{25}{196}=\frac{137}{196}$$

Factor x^{2}-\frac{5}{7}x+\frac{25}{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}.

$$\left(x-\frac{5}{14}\right)^{2}=\frac{137}{196}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{14}\right)^{2}}=\sqrt{\frac{137}{196}}$$

Simplify.

$$x-\frac{5}{14}=\frac{\sqrt{137}}{14} x-\frac{5}{14}=-\frac{\sqrt{137}}{14}$$

Add \frac{5}{14} to both sides of the equation.

$$x=\frac{\sqrt{137}+5}{14} x=\frac{5-\sqrt{137}}{14}$$