Solve 0=(4x^2-29x+60)

$$0 = ( 4 x ^ { 2 } - 29 x + 60 )$$

Solve for x (complex solution)

$$x=\frac{29+\sqrt{119}i}{8}\approx 3.625+1.363589014i$$

$$x=\frac{-\sqrt{119}i+29}{8}\approx 3.625-1.363589014i$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$4x^{2}-29x+60=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -29 for b, and 60 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-29\right)±\sqrt{\left(-29\right)^{2}-4\times 4\times 60}}{2\times 4}$$

Square -29.

$$x=\frac{-\left(-29\right)±\sqrt{841-4\times 4\times 60}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-29\right)±\sqrt{841-16\times 60}}{2\times 4}$$

Multiply -16 times 60.

$$x=\frac{-\left(-29\right)±\sqrt{841-960}}{2\times 4}$$

Add 841 to -960.

$$x=\frac{-\left(-29\right)±\sqrt{-119}}{2\times 4}$$

Take the square root of -119.

$$x=\frac{-\left(-29\right)±\sqrt{119}i}{2\times 4}$$

The opposite of -29 is 29.

$$x=\frac{29±\sqrt{119}i}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{29±\sqrt{119}i}{8}$$

Now solve the equation x=\frac{29±\sqrt{119}i}{8} when ± is plus. Add 29 to i\sqrt{119}.

$$x=\frac{29+\sqrt{119}i}{8}$$

Now solve the equation x=\frac{29±\sqrt{119}i}{8} when ± is minus. Subtract i\sqrt{119} from 29.

$$x=\frac{-\sqrt{119}i+29}{8}$$

The equation is now solved.

$$x=\frac{29+\sqrt{119}i}{8} x=\frac{-\sqrt{119}i+29}{8}$$

Steps for Completing the Square

Swap sides so that all variable terms are on the left hand side.

$$4x^{2}-29x+60=0$$

Subtract 60 from both sides. Anything subtracted from zero gives its negation.

$$4x^{2}-29x=-60$$

Divide both sides by 4.

$$\frac{4x^{2}-29x}{4}=-\frac{60}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}-\frac{29}{4}x=-\frac{60}{4}$$

Divide -60 by 4.

$$x^{2}-\frac{29}{4}x=-15$$

Divide -\frac{29}{4}, the coefficient of the x term, by 2 to get -\frac{29}{8}. Then add the square of -\frac{29}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{29}{4}x+\left(-\frac{29}{8}\right)^{2}=-15+\left(-\frac{29}{8}\right)^{2}$$

Square -\frac{29}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{29}{4}x+\frac{841}{64}=-15+\frac{841}{64}$$

Add -15 to \frac{841}{64}.

$$x^{2}-\frac{29}{4}x+\frac{841}{64}=-\frac{119}{64}$$

Factor x^{2}-\frac{29}{4}x+\frac{841}{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}.

$$\left(x-\frac{29}{8}\right)^{2}=-\frac{119}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{29}{8}\right)^{2}}=\sqrt{-\frac{119}{64}}$$

Simplify.

$$x-\frac{29}{8}=\frac{\sqrt{119}i}{8} x-\frac{29}{8}=-\frac{\sqrt{119}i}{8}$$

Add \frac{29}{8} to both sides of the equation.

$$x=\frac{29+\sqrt{119}i}{8} x=\frac{-\sqrt{119}i+29}{8}$$