Solve 4x^2+7=9x

$$4 x ^ { 2 } + 7 = 9 x$$

Solve for x (complex solution)

$$x=\frac{9+\sqrt{31}i}{8}\approx 1.125+0.695970545i$$

$$x=\frac{-\sqrt{31}i+9}{8}\approx 1.125-0.695970545i$$

Steps Using the Quadratic Formula

Subtract 9x from both sides.

$$4x^{2}+7-9x=0$$

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.

$$4x^{2}-9x+7=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -9 for b, and 7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\times 4\times 7}}{2\times 4}$$

Square -9.

$$x=\frac{-\left(-9\right)±\sqrt{81-4\times 4\times 7}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-9\right)±\sqrt{81-16\times 7}}{2\times 4}$$

Multiply -16 times 7.

$$x=\frac{-\left(-9\right)±\sqrt{81-112}}{2\times 4}$$

Add 81 to -112.

$$x=\frac{-\left(-9\right)±\sqrt{-31}}{2\times 4}$$

Take the square root of -31.

$$x=\frac{-\left(-9\right)±\sqrt{31}i}{2\times 4}$$

The opposite of -9 is 9.

$$x=\frac{9±\sqrt{31}i}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{9±\sqrt{31}i}{8}$$

Now solve the equation x=\frac{9±\sqrt{31}i}{8} when ± is plus. Add 9 to i\sqrt{31}.

$$x=\frac{9+\sqrt{31}i}{8}$$

Now solve the equation x=\frac{9±\sqrt{31}i}{8} when ± is minus. Subtract i\sqrt{31} from 9.

$$x=\frac{-\sqrt{31}i+9}{8}$$

The equation is now solved.

$$x=\frac{9+\sqrt{31}i}{8} x=\frac{-\sqrt{31}i+9}{8}$$

Steps for Completing the Square

Subtract 9x from both sides.

$$4x^{2}+7-9x=0$$

Subtract 7 from both sides. Anything subtracted from zero gives its negation.

$$4x^{2}-9x=-7$$

Divide both sides by 4.

$$\frac{4x^{2}-9x}{4}=-\frac{7}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}-\frac{9}{4}x=-\frac{7}{4}$$

Divide -\frac{9}{4}, the coefficient of the x term, by 2 to get -\frac{9}{8}. Then add the square of -\frac{9}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{9}{4}x+\left(-\frac{9}{8}\right)^{2}=-\frac{7}{4}+\left(-\frac{9}{8}\right)^{2}$$

Square -\frac{9}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{9}{4}x+\frac{81}{64}=-\frac{7}{4}+\frac{81}{64}$$

Add -\frac{7}{4} to \frac{81}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{9}{4}x+\frac{81}{64}=-\frac{31}{64}$$

Factor x^{2}-\frac{9}{4}x+\frac{81}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{9}{8}\right)^{2}=-\frac{31}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{9}{8}\right)^{2}}=\sqrt{-\frac{31}{64}}$$

Simplify.

$$x-\frac{9}{8}=\frac{\sqrt{31}i}{8} x-\frac{9}{8}=-\frac{\sqrt{31}i}{8}$$

Add \frac{9}{8} to both sides of the equation.

$$x=\frac{9+\sqrt{31}i}{8} x=\frac{-\sqrt{31}i+9}{8}$$