Solve 25x^2-40x+23=0

$$25 { x }^{ 2 } -40x+23=0$$

Solve for x (complex solution)

$$x=\frac{4+\sqrt{7}i}{5}\approx 0.8+0.529150262i$$

$$x=\frac{-\sqrt{7}i+4}{5}\approx 0.8-0.529150262i$$

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.

$$25x^{2}-40x+23=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 25 for a, -40 for b, and 23 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-40\right)±\sqrt{\left(-40\right)^{2}-4\times 25\times 23}}{2\times 25}$$

Square -40.

$$x=\frac{-\left(-40\right)±\sqrt{1600-4\times 25\times 23}}{2\times 25}$$

Multiply -4 times 25.

$$x=\frac{-\left(-40\right)±\sqrt{1600-100\times 23}}{2\times 25}$$

Multiply -100 times 23.

$$x=\frac{-\left(-40\right)±\sqrt{1600-2300}}{2\times 25}$$

Add 1600 to -2300.

$$x=\frac{-\left(-40\right)±\sqrt{-700}}{2\times 25}$$

Take the square root of -700.

$$x=\frac{-\left(-40\right)±10\sqrt{7}i}{2\times 25}$$

The opposite of -40 is 40.

$$x=\frac{40±10\sqrt{7}i}{2\times 25}$$

Multiply 2 times 25.

$$x=\frac{40±10\sqrt{7}i}{50}$$

Now solve the equation x=\frac{40±10\sqrt{7}i}{50} when ± is plus. Add 40 to 10i\sqrt{7}.

$$x=\frac{40+10\sqrt{7}i}{50}$$

Divide 40+10i\sqrt{7} by 50.

$$x=\frac{4+\sqrt{7}i}{5}$$

Now solve the equation x=\frac{40±10\sqrt{7}i}{50} when ± is minus. Subtract 10i\sqrt{7} from 40.

$$x=\frac{-10\sqrt{7}i+40}{50}$$

Divide 40-10i\sqrt{7} by 50.

$$x=\frac{-\sqrt{7}i+4}{5}$$

The equation is now solved.

$$x=\frac{4+\sqrt{7}i}{5} x=\frac{-\sqrt{7}i+4}{5}$$

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.

$$25x^{2}-40x+23=0$$

Subtract 23 from both sides of the equation.

$$25x^{2}-40x+23-23=-23$$

Subtracting 23 from itself leaves 0.

$$25x^{2}-40x=-23$$

Divide both sides by 25.

$$\frac{25x^{2}-40x}{25}=-\frac{23}{25}$$

Dividing by 25 undoes the multiplication by 25.

$$x^{2}+\left(-\frac{40}{25}\right)x=-\frac{23}{25}$$

Reduce the fraction \frac{-40}{25} to lowest terms by extracting and canceling out 5.

$$x^{2}-\frac{8}{5}x=-\frac{23}{25}$$

Divide -\frac{8}{5}, the coefficient of the x term, by 2 to get -\frac{4}{5}. Then add the square of -\frac{4}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{8}{5}x+\left(-\frac{4}{5}\right)^{2}=-\frac{23}{25}+\left(-\frac{4}{5}\right)^{2}$$

Square -\frac{4}{5} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{8}{5}x+\frac{16}{25}=\frac{-23+16}{25}$$

Add -\frac{23}{25} to \frac{16}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{8}{5}x+\frac{16}{25}=-\frac{7}{25}$$

Factor x^{2}-\frac{8}{5}x+\frac{16}{25}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{4}{5}\right)^{2}=-\frac{7}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{4}{5}\right)^{2}}=\sqrt{-\frac{7}{25}}$$

Simplify.

$$x-\frac{4}{5}=\frac{\sqrt{7}i}{5} x-\frac{4}{5}=-\frac{\sqrt{7}i}{5}$$

Add \frac{4}{5} to both sides of the equation.

$$x=\frac{4+\sqrt{7}i}{5} x=\frac{-\sqrt{7}i+4}{5}$$