Solve 63x^2-67x+63=0

$$63 { x }^{ 2 } -67x+63=0$$

Solve for x (complex solution)

$$x=\frac{67+\sqrt{11387}i}{126}\approx 0.531746032+0.846903866i$$

$$x=\frac{-\sqrt{11387}i+67}{126}\approx 0.531746032-0.846903866i$$

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.

$$63x^{2}-67x+63=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 63 for a, -67 for b, and 63 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-67\right)±\sqrt{\left(-67\right)^{2}-4\times 63\times 63}}{2\times 63}$$

Square -67.

$$x=\frac{-\left(-67\right)±\sqrt{4489-4\times 63\times 63}}{2\times 63}$$

Multiply -4 times 63.

$$x=\frac{-\left(-67\right)±\sqrt{4489-252\times 63}}{2\times 63}$$

Multiply -252 times 63.

$$x=\frac{-\left(-67\right)±\sqrt{4489-15876}}{2\times 63}$$

Add 4489 to -15876.

$$x=\frac{-\left(-67\right)±\sqrt{-11387}}{2\times 63}$$

Take the square root of -11387.

$$x=\frac{-\left(-67\right)±\sqrt{11387}i}{2\times 63}$$

The opposite of -67 is 67.

$$x=\frac{67±\sqrt{11387}i}{2\times 63}$$

Multiply 2 times 63.

$$x=\frac{67±\sqrt{11387}i}{126}$$

Now solve the equation x=\frac{67±\sqrt{11387}i}{126} when ± is plus. Add 67 to i\sqrt{11387}.

$$x=\frac{67+\sqrt{11387}i}{126}$$

Now solve the equation x=\frac{67±\sqrt{11387}i}{126} when ± is minus. Subtract i\sqrt{11387} from 67.

$$x=\frac{-\sqrt{11387}i+67}{126}$$

The equation is now solved.

$$x=\frac{67+\sqrt{11387}i}{126} x=\frac{-\sqrt{11387}i+67}{126}$$

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.

$$63x^{2}-67x+63=0$$

Subtract 63 from both sides of the equation.

$$63x^{2}-67x+63-63=-63$$

Subtracting 63 from itself leaves 0.

$$63x^{2}-67x=-63$$

Divide both sides by 63.

$$\frac{63x^{2}-67x}{63}=-\frac{63}{63}$$

Dividing by 63 undoes the multiplication by 63.

$$x^{2}-\frac{67}{63}x=-\frac{63}{63}$$

Divide -63 by 63.

$$x^{2}-\frac{67}{63}x=-1$$

Divide -\frac{67}{63}, the coefficient of the x term, by 2 to get -\frac{67}{126}. Then add the square of -\frac{67}{126} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{67}{63}x+\left(-\frac{67}{126}\right)^{2}=-1+\left(-\frac{67}{126}\right)^{2}$$

Square -\frac{67}{126} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{67}{63}x+\frac{4489}{15876}=-1+\frac{4489}{15876}$$

Add -1 to \frac{4489}{15876}.

$$x^{2}-\frac{67}{63}x+\frac{4489}{15876}=-\frac{11387}{15876}$$

Factor x^{2}-\frac{67}{63}x+\frac{4489}{15876}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{67}{126}\right)^{2}=-\frac{11387}{15876}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{67}{126}\right)^{2}}=\sqrt{-\frac{11387}{15876}}$$

Simplify.

$$x-\frac{67}{126}=\frac{\sqrt{11387}i}{126} x-\frac{67}{126}=-\frac{\sqrt{11387}i}{126}$$

Add \frac{67}{126} to both sides of the equation.

$$x=\frac{67+\sqrt{11387}i}{126} x=\frac{-\sqrt{11387}i+67}{126}$$