Solve 14x*x-56=13-2x

$$14x \times x-56=13-2x$$

Solve for x

$$x = \frac{\sqrt{967} - 1}{14} \approx 2.149758829$$

$$x=\frac{-\sqrt{967}-1}{14}\approx -2.292615972$$

Steps Using the Quadratic Formula

Multiply x and x to get x^{2}.

$$14x^{2}-56=13-2x$$

Subtract 13 from both sides.

$$14x^{2}-56-13=-2x$$

Subtract 13 from -56 to get -69.

$$14x^{2}-69=-2x$$

Add 2x to both sides.

$$14x^{2}-69+2x=0$$

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.

$$14x^{2}+2x-69=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 14 for a, 2 for b, and -69 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}-4\times 14\left(-69\right)}}{2\times 14}$$

Square 2.

$$x=\frac{-2±\sqrt{4-4\times 14\left(-69\right)}}{2\times 14}$$

Multiply -4 times 14.

$$x=\frac{-2±\sqrt{4-56\left(-69\right)}}{2\times 14}$$

Multiply -56 times -69.

$$x=\frac{-2±\sqrt{4+3864}}{2\times 14}$$

Add 4 to 3864.

$$x=\frac{-2±\sqrt{3868}}{2\times 14}$$

Take the square root of 3868.

$$x=\frac{-2±2\sqrt{967}}{2\times 14}$$

Multiply 2 times 14.

$$x=\frac{-2±2\sqrt{967}}{28}$$

Now solve the equation x=\frac{-2±2\sqrt{967}}{28} when ± is plus. Add -2 to 2\sqrt{967}.

$$x=\frac{2\sqrt{967}-2}{28}$$

Divide -2+2\sqrt{967} by 28.

$$x=\frac{\sqrt{967}-1}{14}$$

Now solve the equation x=\frac{-2±2\sqrt{967}}{28} when ± is minus. Subtract 2\sqrt{967} from -2.

$$x=\frac{-2\sqrt{967}-2}{28}$$

Divide -2-2\sqrt{967} by 28.

$$x=\frac{-\sqrt{967}-1}{14}$$

The equation is now solved.

$$x=\frac{\sqrt{967}-1}{14} x=\frac{-\sqrt{967}-1}{14}$$

Steps for Completing the Square

Multiply x and x to get x^{2}.

$$14x^{2}-56=13-2x$$

Add 2x to both sides.

$$14x^{2}-56+2x=13$$

Add 56 to both sides.

$$14x^{2}+2x=13+56$$

Add 13 and 56 to get 69.

$$14x^{2}+2x=69$$

Divide both sides by 14.

$$\frac{14x^{2}+2x}{14}=\frac{69}{14}$$

Dividing by 14 undoes the multiplication by 14.

$$x^{2}+\frac{2}{14}x=\frac{69}{14}$$

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

$$x^{2}+\frac{1}{7}x=\frac{69}{14}$$

Divide \frac{1}{7}, the coefficient of the x term, by 2 to get \frac{1}{14}. Then add the square of \frac{1}{14} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{1}{7}x+\left(\frac{1}{14}\right)^{2}=\frac{69}{14}+\left(\frac{1}{14}\right)^{2}$$

Square \frac{1}{14} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{1}{7}x+\frac{1}{196}=\frac{69}{14}+\frac{1}{196}$$

Add \frac{69}{14} to \frac{1}{196} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{1}{7}x+\frac{1}{196}=\frac{967}{196}$$

Factor x^{2}+\frac{1}{7}x+\frac{1}{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}.

$$\left(x+\frac{1}{14}\right)^{2}=\frac{967}{196}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{14}\right)^{2}}=\sqrt{\frac{967}{196}}$$

Simplify.

$$x+\frac{1}{14}=\frac{\sqrt{967}}{14} x+\frac{1}{14}=-\frac{\sqrt{967}}{14}$$

Subtract \frac{1}{14} from both sides of the equation.

$$x=\frac{\sqrt{967}-1}{14} x=\frac{-\sqrt{967}-1}{14}$$