Solve 14x^2-25x+69=125

$$14 { x }^{ 2 } -25x+69=125$$

Solve for x

$$x = \frac{\sqrt{3761} + 25}{28} \approx 3.083106872$$

$$x=\frac{25-\sqrt{3761}}{28}\approx -1.297392587$$

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.

$$14x^{2}-25x+69=125$$

Subtract 125 from both sides of the equation.

$$14x^{2}-25x+69-125=125-125$$

Subtracting 125 from itself leaves 0.

$$14x^{2}-25x+69-125=0$$

Subtract 125 from 69.

$$14x^{2}-25x-56=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 14 for a, -25 for b, and -56 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-25\right)±\sqrt{\left(-25\right)^{2}-4\times 14\left(-56\right)}}{2\times 14}$$

Square -25.

$$x=\frac{-\left(-25\right)±\sqrt{625-4\times 14\left(-56\right)}}{2\times 14}$$

Multiply -4 times 14.

$$x=\frac{-\left(-25\right)±\sqrt{625-56\left(-56\right)}}{2\times 14}$$

Multiply -56 times -56.

$$x=\frac{-\left(-25\right)±\sqrt{625+3136}}{2\times 14}$$

Add 625 to 3136.

$$x=\frac{-\left(-25\right)±\sqrt{3761}}{2\times 14}$$

The opposite of -25 is 25.

$$x=\frac{25±\sqrt{3761}}{2\times 14}$$

Multiply 2 times 14.

$$x=\frac{25±\sqrt{3761}}{28}$$

Now solve the equation x=\frac{25±\sqrt{3761}}{28} when ± is plus. Add 25 to \sqrt{3761}.

$$x=\frac{\sqrt{3761}+25}{28}$$

Now solve the equation x=\frac{25±\sqrt{3761}}{28} when ± is minus. Subtract \sqrt{3761} from 25.

$$x=\frac{25-\sqrt{3761}}{28}$$

The equation is now solved.

$$x=\frac{\sqrt{3761}+25}{28} x=\frac{25-\sqrt{3761}}{28}$$

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.

$$14x^{2}-25x+69=125$$

Subtract 69 from both sides of the equation.

$$14x^{2}-25x+69-69=125-69$$

Subtracting 69 from itself leaves 0.

$$14x^{2}-25x=125-69$$

Subtract 69 from 125.

$$14x^{2}-25x=56$$

Divide both sides by 14.

$$\frac{14x^{2}-25x}{14}=\frac{56}{14}$$

Dividing by 14 undoes the multiplication by 14.

$$x^{2}-\frac{25}{14}x=\frac{56}{14}$$

Divide 56 by 14.

$$x^{2}-\frac{25}{14}x=4$$

Divide -\frac{25}{14}, the coefficient of the x term, by 2 to get -\frac{25}{28}. Then add the square of -\frac{25}{28} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{25}{14}x+\left(-\frac{25}{28}\right)^{2}=4+\left(-\frac{25}{28}\right)^{2}$$

Square -\frac{25}{28} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{25}{14}x+\frac{625}{784}=4+\frac{625}{784}$$

Add 4 to \frac{625}{784}.

$$x^{2}-\frac{25}{14}x+\frac{625}{784}=\frac{3761}{784}$$

Factor x^{2}-\frac{25}{14}x+\frac{625}{784}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{25}{28}\right)^{2}=\frac{3761}{784}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{25}{28}\right)^{2}}=\sqrt{\frac{3761}{784}}$$

Simplify.

$$x-\frac{25}{28}=\frac{\sqrt{3761}}{28} x-\frac{25}{28}=-\frac{\sqrt{3761}}{28}$$

Add \frac{25}{28} to both sides of the equation.

$$x=\frac{\sqrt{3761}+25}{28} x=\frac{25-\sqrt{3761}}{28}$$