Solve 4x^2=10x-9

$$4 x ^ { 2 } = 10 x - 9$$

Solve for x (complex solution)

$$x=\frac{5+\sqrt{11}i}{4}\approx 1.25+0.829156198i$$

$$x=\frac{-\sqrt{11}i+5}{4}\approx 1.25-0.829156198i$$

Steps Using the Quadratic Formula

Subtract 10x from both sides.

$$4x^{2}-10x=-9$$

Add 9 to both sides.

$$4x^{2}-10x+9=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -10 for b, and 9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\times 4\times 9}}{2\times 4}$$

Square -10.

$$x=\frac{-\left(-10\right)±\sqrt{100-4\times 4\times 9}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-10\right)±\sqrt{100-16\times 9}}{2\times 4}$$

Multiply -16 times 9.

$$x=\frac{-\left(-10\right)±\sqrt{100-144}}{2\times 4}$$

Add 100 to -144.

$$x=\frac{-\left(-10\right)±\sqrt{-44}}{2\times 4}$$

Take the square root of -44.

$$x=\frac{-\left(-10\right)±2\sqrt{11}i}{2\times 4}$$

The opposite of -10 is 10.

$$x=\frac{10±2\sqrt{11}i}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{10±2\sqrt{11}i}{8}$$

Now solve the equation x=\frac{10±2\sqrt{11}i}{8} when ± is plus. Add 10 to 2i\sqrt{11}.

$$x=\frac{10+2\sqrt{11}i}{8}$$

Divide 10+2i\sqrt{11} by 8.

$$x=\frac{5+\sqrt{11}i}{4}$$

Now solve the equation x=\frac{10±2\sqrt{11}i}{8} when ± is minus. Subtract 2i\sqrt{11} from 10.

$$x=\frac{-2\sqrt{11}i+10}{8}$$

Divide 10-2i\sqrt{11} by 8.

$$x=\frac{-\sqrt{11}i+5}{4}$$

The equation is now solved.

$$x=\frac{5+\sqrt{11}i}{4} x=\frac{-\sqrt{11}i+5}{4}$$

Steps for Completing the Square

Subtract 10x from both sides.

$$4x^{2}-10x=-9$$

Divide both sides by 4.

$$\frac{4x^{2}-10x}{4}=-\frac{9}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\left(-\frac{10}{4}\right)x=-\frac{9}{4}$$

Reduce the fraction \frac{-10}{4} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{5}{2}x=-\frac{9}{4}$$

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

$$x^{2}-\frac{5}{2}x+\left(-\frac{5}{4}\right)^{2}=-\frac{9}{4}+\left(-\frac{5}{4}\right)^{2}$$

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

$$x^{2}-\frac{5}{2}x+\frac{25}{16}=-\frac{9}{4}+\frac{25}{16}$$

Add -\frac{9}{4} to \frac{25}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{5}{2}x+\frac{25}{16}=-\frac{11}{16}$$

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

$$\left(x-\frac{5}{4}\right)^{2}=-\frac{11}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{4}\right)^{2}}=\sqrt{-\frac{11}{16}}$$

Simplify.

$$x-\frac{5}{4}=\frac{\sqrt{11}i}{4} x-\frac{5}{4}=-\frac{\sqrt{11}i}{4}$$

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

$$x=\frac{5+\sqrt{11}i}{4} x=\frac{-\sqrt{11}i+5}{4}$$