Solve left(5xright)^2-7x+2=0

$${ \left(5x \right) }^{ 2 } -7x+2=0$$

Solve for x (complex solution)

$$x=\frac{7+\sqrt{151}i}{50}\approx 0.14+0.245764115i$$

$$x=\frac{-\sqrt{151}i+7}{50}\approx 0.14-0.245764115i$$

Steps Using the Quadratic Formula

Expand \left(5x\right)^{2}.

$$5^{2}x^{2}-7x+2=0$$

Calculate 5 to the power of 2 and get 25.

$$25x^{2}-7x+2=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 25 for a, -7 for b, and 2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Square -7.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\times 25\times 2}}{2\times 25}$$

Multiply -4 times 25.

$$x=\frac{-\left(-7\right)±\sqrt{49-100\times 2}}{2\times 25}$$

Multiply -100 times 2.

$$x=\frac{-\left(-7\right)±\sqrt{49-200}}{2\times 25}$$

Add 49 to -200.

$$x=\frac{-\left(-7\right)±\sqrt{-151}}{2\times 25}$$

Take the square root of -151.

$$x=\frac{-\left(-7\right)±\sqrt{151}i}{2\times 25}$$

The opposite of -7 is 7.

$$x=\frac{7±\sqrt{151}i}{2\times 25}$$

Multiply 2 times 25.

$$x=\frac{7±\sqrt{151}i}{50}$$

Now solve the equation x=\frac{7±\sqrt{151}i}{50} when ± is plus. Add 7 to i\sqrt{151}.

$$x=\frac{7+\sqrt{151}i}{50}$$

Now solve the equation x=\frac{7±\sqrt{151}i}{50} when ± is minus. Subtract i\sqrt{151} from 7.

$$x=\frac{-\sqrt{151}i+7}{50}$$

The equation is now solved.

$$x=\frac{7+\sqrt{151}i}{50} x=\frac{-\sqrt{151}i+7}{50}$$

Steps for Completing the Square

Expand \left(5x\right)^{2}.

$$5^{2}x^{2}-7x+2=0$$

Calculate 5 to the power of 2 and get 25.

$$25x^{2}-7x+2=0$$

Subtract 2 from both sides. Anything subtracted from zero gives its negation.

$$25x^{2}-7x=-2$$

Divide both sides by 25.

$$\frac{25x^{2}-7x}{25}=-\frac{2}{25}$$

Dividing by 25 undoes the multiplication by 25.

$$x^{2}-\frac{7}{25}x=-\frac{2}{25}$$

Divide -\frac{7}{25}, the coefficient of the x term, by 2 to get -\frac{7}{50}. Then add the square of -\frac{7}{50} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{7}{25}x+\left(-\frac{7}{50}\right)^{2}=-\frac{2}{25}+\left(-\frac{7}{50}\right)^{2}$$

Square -\frac{7}{50} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{7}{25}x+\frac{49}{2500}=-\frac{2}{25}+\frac{49}{2500}$$

Add -\frac{2}{25} to \frac{49}{2500} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{7}{25}x+\frac{49}{2500}=-\frac{151}{2500}$$

Factor x^{2}-\frac{7}{25}x+\frac{49}{2500}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{7}{50}\right)^{2}=-\frac{151}{2500}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7}{50}\right)^{2}}=\sqrt{-\frac{151}{2500}}$$

Simplify.

$$x-\frac{7}{50}=\frac{\sqrt{151}i}{50} x-\frac{7}{50}=-\frac{\sqrt{151}i}{50}$$

Add \frac{7}{50} to both sides of the equation.

$$x=\frac{7+\sqrt{151}i}{50} x=\frac{-\sqrt{151}i+7}{50}$$